Polarons and Polaritons in Cesium Lead Bromide Perovskite Tyler James Swenson Evans

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy under the Executive Committee

of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2018

ã 2018 Tyler James Swenson Evans

All rights reserved

ABSTRACT

Polarons and Polaritons in Cesium Lead Bromide Perovskite

Tyler James Swenson Evans

Lead halide perovskites are a class of soft ionic semiconductors characterized by strong

excitonic absorption and long carrier lifetimes. Recent studies suggest that electrons and holes

in these materials interact with longitudinal optical phonons to form large polarons on

subpicosecond time-scales. The same interaction is responsible for hot electron cooling via

phonon emission and is thought to be screened by large polaron formation resulting in the

long-lived hot electrons observed in methylammonium lead iodide perovskite. Time-resolved

two-photon photoemission is used to follow the initial hot electron cooling and large polaron

formation dynamics in single-crystal cesium lead bromide perovskite at 80 K and 300 K. The

initial relaxation rates are found to be weakly temperature-dependent and are attributed to the

cooling of unscreened hot electrons by the emission of longitudinal optical phonons. The large

polaron formation times, however, are inferred to be approximately three times faster at 300

K. The decrease in polaron formation time with temperature is correlated with the broadening

in phonon linewidths, suggesting that disorder can assist large polaron formation. In addition,

the initial electron relaxation is faster than large polaron formation explaining the absence of

long-lived hot electrons in cesium lead bromide perovskite as opposed to methylammonium

lead iodide perovskite where the two processes are competitive. The second part of this thesis

focuses on the strong light-matter interaction in nanowire waveguide geometries of single-

crystal lead halide perovskites which are well known for their emission tunability and low lasing

thresholds under pulsed optical excitation. Using fluorescence microscopy, it is found that the

luminescence from single-crystal cesium lead bromide perovskite nanowires is dominated by

sub-bandgap modes called exciton-polaritons, i.e. hybridized exciton-photon states. A one-

dimensional exciton-polariton model reproduces the observed modes at the bottleneck of the

lower polariton branch with a Rabi splitting of about 200 milli-electron volts. As the power

density increases under continuous excitation, the exciton-polaritons undergo Bose-stimulated

scattering and a super-linear increase in mode intensity is observed. This is the first

demonstration of continuous-wave lasing in lead halide perovskite nanowires and reveals an

inherently strong light-matter interaction in lead halide perovskites that can be used for

continuous-wave optoelectronic applications. These findings corroborate the role of dynamic

screening in unifying these two regimes of carrier-carrier interactions responsible for the

strong absorption and subsequent carrier protection. We also demonstrate the viability of lead

halide perovskite nanowires for future optoelectronics.

i

Contents

List of Figures iii

1 Motivation 1

2 Lead Halide Perovskites 4

2.1 Coherent Photoexcitation and Dephasing 10

2.2 Thermalization Processes and Hot Carriers 15

3 Large Polaron Screening and Hot-Electron Cooling in Bulk CsPbBr3 18

3.1 CsPbBr3 Sample Preparation 19

3.2 Time-Resolved Two-Photon Photoemission Spectroscopy 20

3.2 Results and Discussion 21

4 Bose-Stimulated Scattering of Exciton-Polaritons in CsPbBr3 Nanowires 35

4.1 Results and Discussion 39

4.2 Conclusions 49

5 Concluding Remarks 53

ii

References 54

iii

List of Figures

Figure 2.1 Cubic, Tetragonal, and Orthorhombic Perovskite Structures 5

Figure 2.2 Soft Modes and Phonon Bandstructure from HELIX 6

Figure 2.3 Ferroelectric-like Far Infrared Transmittance Spectrum of LHPs 7

Figure 2.4 SOC-GW Bandstructure Calculations of CH3NH3PbI3 8

Figure 2.5 Optical Absorption Spectrum and Elliot Deconvolution 9

Figure 2.6 Two-Dimensional Electronic Spectra and Exciton Screening 11

Figure 2.7 Ultrafast Dielectric Response of Electron-Hole Plasma in GaAs 12

Figure 2.8 Exciton-Polaritons in CsPbCl3 Microcavities 13

Figure 2.9 Carrier Thermalization Rates with Excess Energy 14

Figure 2.10 Optical Phonons in Transient Transmission Spectra of LHPs 15

Figure 2.11 Transient Photoluminescence of FAPbBr3 and CsPbBr3 16

Figure 3.1 Electron Distribution Dynamics in Single Crystal CsPbBr3 22

Figure 3.2 Population Dynamics in Single Crystal CsPbBr3 25

Figure 3.3 Disorder Assisted Large Polaron Screening 30

Figure 4.1 Exciton-Polariton Model and Dispersion 37

Figure 4.2 AFM Images of CsPbBr3 Nanowires 39

Figure 4.3 Power Dependence and Lasing 41

Figure 4.4 Nonlinear Modes and Subbandgap Luminescence 43

iv

Figure 4.5 Lower Polariton Dispersion Fits and Rabi Splittings 45

Figure 4.6 Complete Exciton-Polariton Model for CsPbBr3 48

Figure 4.7 Temperature Dependence of Pulsed Lasing 50

1

Motivation 1

Lead halide perovskites (LHPs), APbX3 (A = molecular or alkali metal cation; X = Cl, Br,

I), are soft, polar semiconductors renowned for their high efficiencies as photovoltaics and

light-emitting devices.1–6 Many properties essential to device performance–charge carrier

lifetimes, mobilities, and recombination rates–in LHPs have been unified in the large polaron

model which assumes that electrons and holes are dynamically screened by the polar and soft

crystalline lattice.7–9 The efficient screening diminishes the Coulomb potential of the resulting

large polaron and its scattering with other electrons/holes, trapped charges, and additional LO

phonons potentially leading to hot carrier distributions as seen in hybrid LHPs.10,11 Favorable

defect chemistry and screening of charged defects can greatly reduce the temperature at which

the carrier mobilities become limited by defect scattering, thus explaining the defect tolerance

observed in LHPs with densities of ~ 1015 – 1017 cm-3.12–14 The response time of the LO

phonon interaction initially leaves the carriers unscreened giving rise to strong excitonic

absorption in LHPs despite negligible exciton population.15–17 The disparate behavior of LHPs

before and after large polaron formation give rise to high photoluminescence quantum

efficiencies, low amplification thresholds, and high nonlinear optical coefficients desirable for

next-generation, solution-processed optoelectronics.18–24

Despite the fundamental role of the large polaron to LHP technology, its nature is

elusive due to the soft and disordered character of the phonon modes.9,25–29,8 In the classic

2

Fröhlich model, electrons and holes interact with harmonic LO phonons that depend linearly

on the LO phonon displacement.30 The modes in LHPs deviate drastically from these

assumptions and share the characteristics of relaxor ferroelectrics.29,31,32 This has led to the

proposal by Miyata et al. of a ferroelectric large polaron in which the electric field of a charge

carrier can induce a local ferroelectric phase transition and ordering of nearby polar regions,

screening the Coulomb potential of the carrier even further.31 The effects of polar fluctuations

on the formation of this new type of large polaron is vastly unexplored despite the potential

to apply this mechanism for carrier protection in future nanostructured materials.8,31 In

Chapter 4 of this thesis we probe; 1) the bare electron-phonon interaction, before dielectric

relaxation, and 2) the effect of polar fluctuations on large polaron formation by comparing

room-temperature dynamics to those at 80 K. The original research goal was to warm CsPbBr3

out of the low symmetry orthorhombic phase and into the higher symmetry tetragonal and

cubic phases. This would allow for direct comparison with CH3NH3PbI3 and CH3NH3PbBr3

to determine whether the lack of hot-carriers (discussed in Chapter 2) in CsPbBr3 was related

to the lack of an organic cation or simply due to the structural phase. It turned out to be more

viable if one stayed within the same structural phase and simply tuned the dynamic disorder.

By doing this, we learned the more fundamental lesson that large polaron formation is assisted

by dynamic disorder and the presence of hot carriers is determined by the competition

between carrier cooling and dielectric response of the LO phonons. Tuning this response in

nanostructured materials could lead to advancements in hot carrier based technologies.

The initial, unscreened Coulomb interaction in LHPs gives rise to their superior optical

properties.15–17 Lead halide perovskite nanowires (NWs) have demonstrated pulsed lasing with

3

high quantum yields, low thresholds, and broad tunability.33–37 The all-inorganic LHPs, in

which the A site cation is Cs+, are more robust than their hybrid organic−inorganic

counterparts and have drawn attention as robust materials for efficient light-

emission.34,36,38,39 However, continuous-wave (CW) lasing, necessary for many optoelectronic

applications, had not yet been achieved. It was thought to be due to many-body screening,

which reduces the excitonic resonance enhancement of the oscillator strength at high

excitation densities necessary for population inversion. In Chapter 5, CW lasing in CsPbBr3

perovskite NWs is reported. Analysis of the cavity modes and their temperature dependence

reveals that CW lasing originate from polariton modes near the bottleneck region on the lower

polariton branch, with a vacuum Rabi splitting of 0.20 ± 0.03 eV. These findings suggest that

lead halide perovskite NWs may serve as low-power CW coherent light sources and as model

systems for polaritonics in the strong-coupling regime.

Given the extensive body of LHP research a review of the literature will be presented

in Chapter 2 in attempt to present a cogent argument for dynamic screening and large polaron

formation in LHPs. This will include an overview of dynamic disorder in LHPs, electronic

structure, and ultrafast studies of carrier dynamics. I will also make a connection to polaritonics

in the context of dynamic screening. This will lead up to Chapter 4 and Chapter 5 on large

polarons and exciton-polaritons, respectively.

4

Lead Halide Perovskites 2

LHPs are composed of an inorganic PbX6-4 (X = Cl, Br, I) corner-sharing octahedral network

with interstitial cations A [A = Rb+, Cs+, CH3NH3+ (MA), CH(NH2)2

+ (FA), C(NH2)3+ (GA)]

that may be organic or inorganic in nature, APbX3.4,40,41 In the ideal cubic structure, the X-Pb-

X bond angles are 90° with the A site cation sitting in a cubic cage.42 However, the mismatch

in ionic radii between the halide and the A site cation leads to structural instabilities resulting

in the twisting and tilting of octahedra as shown in Figure 2.1.42–44 In fact, the ideal cubic phase

does not exist in LHPs, but rather manifests as an average over lower symmetry domains

shown by x-ray pair distribution functions of single-crystal CH3NH3PbI3 to be on the order

of a few nanometers.27 High energy resolution x-ray scattering on CH3NH3PbI3 in the cubic

phase reveals that the dynamically unstable tilting modes are highly populated and soften

towards the zone edge due to large anharmonicities.27 These soft modes are identified as the

in- and out of- phase tilting modes and are shown in Figure 2.2. As the crystal is cooled, the

cubic to tetragonal and the tetragonal to orthorhombic transitions occur by the condensation

of the out of- and in- phase modes, respectively.27 Low-frequency Raman spectroscopy reveals

that the soft mode behavior is general to all LHPs and not unique to only hybrid LHPs.45

Molecular dynamics and density functional theory calculations confirm a displacement of the

A site cation accompanying octahedral tilting confirming the polar nature of the modes.29,44

This behavior is featured broadly in a class of materials called relaxor ferroelectrics where long

range interactions produce polar nanoregions that are dynamically disordered.31,46,47 The low

5

frequencies and large anharmonicities can also account for the dismal thermal conductivities

of these materials.48–50

Evidence of ferroelectric like behavior is found in the far infrared reflectance spectrum

of CH3NH3PbX3, (X = Cl, Br, I), shown in Figure 2.3. In this region the transverse optical

(TO) and longitudinal optical (LO) modes are probed. The spectrum reveals a large LO-TO

splitting and highly anharmonic modes.28 In an analysis by Miyata et al., it was shown that the

real part of the dielectric function is increased by an order of magnitude beyond the TO

resonance, compared to GaAs which possesses a less than 20% increase, and is strikingly

similar to the ferroelectric material SrBi2Ta2O9.31 The large enhancements of the dielectric

constant is due to the large phonon anharmonicity in LHP and ferroelectrics near a phase

transition, reflected by the TO mode linewidths, that allow for a higher degree of lattice

(a) (b)

(c)

Figure 2.1: The three structural phases of lead halide perovskites; (a) orthorhombic, (b)

tetragonal, and (c) cubic. Note the out of phase twist in (b) versus the in phase twist in

(a). Figure adapted from Brivio et al.25

6

polarizability.31 The soft, anharmonic modes present a significant challenge to our

understanding of the carrier dynamics in these materials which are driven largely by

interactions of electrons and holes with phonons.51

How electrons and holes interact with phonons in a given material greatly depends

on the electron structure.52 Due to the initial, unscreened interaction the electronic structure

will also determine the high frequency optical response of the material.16,17,53 The bandstructure

calculated for CH3NH3PbI3 is shown in Figure 2.4 and includes spin-orbit coupling (SOC) and

Figure 2.2: (a) The in phase and out of phase octahedral twist modes MTA1 and RTA,

respectively. (b) Experimental acoustic phonon dispersion measured by high energy

resolution x-ray scattering showing the soft mode behavior of the modes in (a). Note that

the A site cation displacements are not considered in this cartoon. Figure adapted from

Beecher et al.27

(a)

(b)

7

many-body interactions with valence electrons.53 In LHPs generally, the X 5p orbitals and Pb

6s orbitals hybridize to form the valence band while the conduction states form mainly from

Pb 6p orbitals, in contrast to III-V and II-VI materials where the conduction states derive

from s orbitals and the valence states from p orbitals.53–55 The A site cations can affect the

inorganic sub-lattice equilibrium positions and dynamics based on the size and rotational-

vibrational motions but they do not hybridize with the lattice and therefore do not contribute

electronically to the bandedge states.56–60 The heavy lead nuclei are responsible for the

relativistic speeds of electrons in the Pb 6s orbitals that contribute to the valence band, along

with X 5p, and could contribute to the small high frequency dielectric constant of LHPs that

Figure 2.3: The far infrared transmittance spectrum of CH3NH3PbI3 is shown for both

normal (top) and 70° incidence angle. At 70° incidence angle there is absorption by the LO

mode. Note that the resonances are broader for CH3NH3PbI3 than those for PbI2. Figure

adapted from Sendner et al.28

8

ranges from about 4 to 5 varying with halide polarizability–from Cl to I–compared to a value

of 10 for GaAs.28 This reveals a stark contrast between the two materials: GaAs is about ten

times harder than CH3NH3PbI3 and the static dielectric constant is largely due to the electronic

screening whereas in LHPs, the screening is predominantly from lower frequency

responses.27,28 A large SOC among the Pb 6p orbitals lifts the degeneracy of the Pb derived

conduction bands effectively lowering the conduction band minimum (CBM) while increasing

the electronic bandwidth to give reduced and more isotropic effective masses.53,61 The bandgap

of LHPs is tunable and ranges from about 1.6 to 3.2 eV varying with strongly with X–from I

to Cl–and the subtly on the size of A (only reporting for CH3NH3PbI3 and Cs).4,34

The large spin-orbit interaction of the Pb 6p orbitals lifts the degeneracy of the

conduction bands effectively reducing the bandgap and increases the effective mass isotropy.62

The light effective band masses highlights the tendency of carriers to delocalize from

(a) (b)

Figure 2.4: (a) The SOC-GW bandstructure of CH3NH3PbI3 in the SR-DFT relaxed

structure shown in (b). Note the out of phase octahedral tilting characteristic of the

tetragonal phase of room temperature CH3NH3PbI3. Figure adapted from Umari et al.53

9

uncertainty arguments in the effective band approximation, 𝛥𝑥 > ℏ 2𝑚𝛥𝐸 )*/,, where 𝛥𝑥

is the spread in the electron wavefunction, ℏ is Planck’s constant, 𝑚 is the effective mass and

𝛥𝐸 is the electronic bandwidth. This will counter act the tendency of electrons and holes to

localize in LHPs from dynamic disorder and electron-phonon interactions.52 Effective masses

give a qualitative picture on optical properties as the coupling between valence and conduction

states, which is proportional to the oscillator strength, scales inversely with the effective mass,

𝑣 𝑝 𝑐 , ∝ 𝑚)*, where |𝑣 and |𝑐 are the valence and conduction band states, respectively,

and 𝑝 is the single-particle momentum operator.63 Despite the decrease in the joint density of

states from the light effective masses and conduction band splitting, the absorption

coefficients of LHPs are comparable to GaAs (~105 cm-1).64,65 The absorption spectrum of

Figure 2.5: Absorption spectrum of a CH3NH3PbI3 thin film at room temperature (red

circles). The data is fit using an Elliot formula which accounts for the Coulomb interaction

between electrons and holes that give rise to discretized exciton bound states. Figure

adapted from Yang et al.15

10

LHPs is described by an interband polarization continuum with enhanced oscillator strength,

relative to free-carrier transitions, due to strong electron-hole Coulomb interactions.15,16,66 This

is captured by the Elliot model in which electron and hole interactions are included.66 Below

the onset of the continuum absorption are the bound electron-hole states or excitons.66 A

single exciton resonance is observed in LHPs due to the low, ~10 meV, exciton binding energy

shown in Figure 2.5.15,16,67

Given that we now have some understanding of the complexity of the ground state

nuclear motions in LHPs, the atomistic origins of the conduction and valence bands, and the

excitonic linear absorption spectrum, we now turn our discussion to the dynamics of large

polaron formation, beginning with photoexcitation, to present the current understanding of

dynamic screening in LHPs.

2.1 Coherent Photoexcitation and Dephasing

Upon photoexcitation, a coherent wavepacket is generated that may consist of both

continuum and bound states depending on the photon energies used in the experiment that is

instantaneously screened by the valence electrons.68 The initial coherence or polarization is

observed in, for example, linear absorption spectroscopy which probes the distributed energies

of coherences and is broadened by their dephasing rates and energetic inhomogeneities caused

by the surroundings.68 Ultrafast two-dimensional electronic spectroscopy (2DES) is a powerful

tool for studying electronic coherences that allows one to directly measure the dephasing times

and mechanisms providing insight into intrinsic scattering processes in the material.69

Experiments on this time-scale are few. Jha et al. successfully recorded 2DES of MAPbI3 with

11

sub-20 fs timeresolution performed in the tetragonal crystal phase with carrier densities from

approximately 1018 cm-3 to 1019 cm-3.17 In this experiment, the exciton resonance and

continuum absorption are resolved revealing electronic dephasing rates of ~ 50 fs with no

resulting exciton population.17 A temperature and power dependence determined that the

dephasing mechanism is electronic in nature and is due to the bare Coulomb interaction, 𝑉𝒒,

between carriers, where 𝒒 is the momentum exchange between two interacting carriers68. This

is not surprising considering the expected delayed response of the soft modes and the relatively

high carrier densities that lead to electronic responses on the order of the plasma frequency,

𝜔5 = 789

:;<

*/,, where 𝑛 is the carrier density, 𝑒, is the elementary charge and 𝜀∞ is the

high-frequency dielectric constant.63,68 The exciton resonance vanishes on the same dephasing

timescale as shown in Figure 2.6, suggesting that there is a screening effect possibly due to a

newly formed electron-hole plasma as suggested by the relationship between Coulomb

Figure 2.6: Two-dimensional electronic spectra of CH3NH3PbI3 at near band-edge

excitation after (a) 0 fs and (b) 50 fs. The excitonic feature labeled A in (a) is screened by

the population of electrons and holes that form within 50 fs. Figure adapted from Jha et

al.17

12

scattering and plasma formation.68 To be clear, the plasma screening discussed here is different

from the large polaron screening. Plasma screening occurs when electrons and holes reorient

themselves spatially to effectively reduce the single-particle Coulomb potential of any one

charge.68,70,71 Huber et al. demonstrated the validity of the many-body concept of a time-

dependent dielectric function by measuring the ultrafast dynamics of plasma formation

following photoexcitation in GaAs.68 It was that he dielectric response of the plasma is delayed

on the order of 102 fs, approximately the period of plasma oscillation. This is captured

physically by a time-dependent effective Coulomb interaction potential for the carriers,

𝑊𝒒 𝜔, 𝑡 = 𝜀𝒒)𝟏 𝜔, 𝑡 𝑉𝒒, where 𝜀𝒒 is the time-dependent longitudinal dielectric function. The

real and imaginary parts of the dielectric response are measured by a terahertz probe and

Figure 2.7: Ultrafast formation of electron hole plasma resonance shown by the imaginary

part of the dielectric function (a) and the parallel buildup of plasma screening shown by

the real part of the dielectric function (b). Figure adapted from Huber et al.71

(a) (b)

13

shown in Figure 2.7. Though distinct from the large polaron, the same type of dynamic

dielectric screening can be used to describe large polaron formation: remember, the majority

of the dielectric response in GaAs is electronic, but in LHPs its twice as large and mainly

nuclear. The takeaway from this section is that photoexcited coherences dephase rapidly in 50

fs due to weakly screened Coulomb interaction resulting in exciton screening on the same

timescale.17,72

Figure 2.8: Angle-resolved photoluminescence spectra from a CsPbCl3 microcavity. As

the condensation or lasing threshold is approached the emission narrows in momentum

space. Example of unscreened interactions. Figure adapted from Su et al.73

14

The strong carrier-carrier interactions in the coherence regime lead to greater electron-

hole wavefunction overlap and enhanced oscillator strengths that have major implications for

optoelectronics as this can be exploited for strong nonlinear optical interactions.4 The exciton

oscillator strength is dependent on the transition dipole moment, 𝜇EF ≡ 0 𝜇 𝑋 , between

the ground, |0 , and exciton state, |𝑋 . This value has been reported by Yang, Y. et al. who

measured a linear optical stark shift in MAPbI3 that is consistent with a transition dipole

moment of 𝜇EF ~ 46 D or an oscillator strength of ~ 38, shockingly similar to that found in

quantum-confined 2D MoS2.15 Naturally, this has led to strong coupling effects resulting in

Rabi splittings of 265 meV directly observed in the angle-resolved photoluminescence spectra

of CsPbCl3 microcavities shown in Figure 2.8.73 Measured by Su, R. et al., these systems were

Figure 2.9: Dependence of the thermalization times and scattering rates on the excess

energy (ℏ𝜔 − 𝐸K) at 1018 cm-3 (red circles) and 1019 cm-3 (black squares). As the

condensation or lasing threshold is approached the emission narrows in momentum space.

Example of unscreened interactions. Figure adapted from Richter et al.76

15

also shown to support polariton lasing and long-range (~ 10 𝜇m) spatial coherence.73 This is

directly related to the great success of perovskite nanowire lasers and other

morphologies.33,36,74,75

2.2 Thermalization Processes and Hot Carriers

Due to the softness of the modes in LHPs, carrier-LO phonon scattering will outpace the

plasma response for densities less than roughly 1017 cm-3, an order of magnitude lower than in

GaAs.71 After dephasing, the population distribution at 50 fs is highly nonequilibrium and is

related to the initial coupling of light and subsequent relaxation to coupled states described by

the optical Bloch equations.68 This distribution begins to interact with the environment and

undergo momentum and energy relaxation.63,68 Due to the slow response of the phonons, the

electrons and holes will initially interact and acquire a temperature that is higher than the lattice

Figure 2.10: Two-dimensional Fourier map for (a) CH3NH3PbBr3 and (b) CH3NH3PbI3

showing previously unresolved optical phonon modes at (a) 92 cm-1 and (b) 110 cm-1

appearing on either side of the exciton resonance. Figure adapted from Ghosh et al.72

16

temperature.63 Richter et al. used 2DES to probe the thermalization dynamics in CH3NH3PbI3

films directly by observing the population relaxation after above bandgap photoexcitation with

sub-10 fs time-resolution.76 Performing these measurements and different photon energies

which keeping the carrier densities pinned at 1018 cm-3 and 1019 cm-3 they report the

thermalization times in Figure 2.9.76 This timescale is also reported by Ghosh et al. who

reported thermalization within 102 fs.72

The thermalization of electrons and holes at 102 fs is followed by the thermalization

of electrons and holes with the optical phonons of the material on the picosecond

timescale.63,77,78 Impulsive coherent phonons where measured by Ghosh et al. in pump-probe

measurements whose Fourier transforms are shown in Figure 2.10.72 The data shows strong

electron-phonon coupling which is expected in a material with such a large nuclear

polarizability, as discussed in the beginning of the chapter. The cooling of the hot electron

distribution by equilibration with the optical modes is a complicated process in LHPs.10,11,79,80

One complication is the presence of long-lived hot carriers–102 ps–that are observed in the

Figure 2.11: Time-resolved photoluminescence of (a) CH(NH2)2PbBr3 and (b) CsPbBr3.

Note the high energy PL lasting for 102 ps. Figure adapted from Zhu et al.10

17

time-resolved photoluminescence of hybrid LHPs shown in Figure 2.11.10 The mechanism

responsible for LO phonon emission, the Fröhlich interaction, is a long-range Coulomb

interaction between the carrier and a deformable and polarizable lattice.52 Given the nature of

the modes that we covered in this chapter, screening by the ordering of the polaron

nanoregions, as suggested by Miyata et al., could screen the Coulomb potential of the carrier

sufficiently to drastically reduce the LO phonon scattering times.31 It has also been shown to

be possible that the large polaron induced strain could lead to repulsive forces between

oppositely charged large polarons.81

18

Large Polaron Screening and Hot Electron Cooling 4

Understanding the nature and formation of large polarons in LHPs is not only crucial to LHP

based technology but it provides a path to overcoming the transport problems common to

self-assembled semiconductors. Miyata et al. recently probed polaron formation dynamics in

LHPs using time-resolved optical Kerr effect spectroscopy (TR-OKE) to follow the phonon

response upon charge injection. This measurement yielded room-temperature polaron

formation time constants of τ = 0.3 ps in MAPbBr3 and 0.7 ps in CsPbBr3.9 The faster polaron

formation in hybrid organic−inorganic LHPs is correlated with the presence of long-lived hot

carriers, not present in CsPbBr3,10,11 suggesting that large polaron screening may contribute to

the slowed cooling of hot carriers. Since the interaction of a nascent electron with LO phonons

is responsible for both initial hot electron cooling and large polaron formation,30,82,52 how the

competition between these two processes ultimately determines the fate of energetic

carriers10,11 remains ambiguous.

Using TR-2PPE spectroscopy, we directly probe the electron distribution in the

conduction band of single-crystal CsPbBr3 following above-bandgap photoexcitation at both

room and liquid nitrogen temperatures. Note that, at both temperatures, CsPbBr3 is in the

low- temperature orthorhombic phase.83 We determine the initial (≤0.2 ps) energy relaxation

rates of nascent hot electrons of 𝛾M = −0.64 ± 0.06 eV ps-1 and −0.82 ± 0.08 eV ps-1 at 300

and 80 K, respectively. This weakly temperature-dependent relaxation rate is expected from

19

the cooling of unscreened hot electrons by LO phonon emission, with an electron-LO phonon

scattering time constant of 𝜏8)OP = 3.8 ± 0.5 fs. In contrast to the weak temperature-

dependence in hot electron cooling, we find that polaron formation occurring on longer times

is strongly temperature dependent, with time constants increasing from 𝜏5 = 0.7 ± 0.1 ps at

300 K to 2.2 ± 0.3 ps at 80 K. Here 𝜏5 is theoretically related to the buildup of

electron−phonon correlation.84 We can attribute faster polaron formation at higher

temperatures to higher phonon−phonon scattering rates and, thus shorter phonon lifetimes

and broader phonon resonances. Our findings are consistent with known low-frequency

Raman spectra of LHPs,29,60 and corroborate the crystal−liquid duality of LHPs.8

3.1 CsPbBr3 Single Crystals

We grew single-crystals of CsPbBr3 under ambient conditions by vapor diffusion

whereby antisolvent vapor diffuses into a precursor solution to induced

crystallization.85 The precursor was an 0.4 M solution of PbBr2 and CsBr (1:1 molar

ratio) in dimethyl sulfoxide with methanol as an antisolvent. Details on crystal growth

and characterization have been reported elsewhere.9 Single-crystal samples with lateral

sizes of a few millimeters were mounted on a native-oxide terminated Si(111) wafer

with Ag epoxy and loaded into a vacuum system for laser spectroscopic measurements.

Each crystal was freshly cleaved in an ultrahigh vacuum (UHV) sample preparation

chamber (<5 × 10−10 mbar) and moved to an UHV analysis chamber (<5 × 10−11

mbar) for in TR-2PPE measurement. Alternatively, each crystal was mounted in a

vacuum cryostat (< 10−6 mbar) for TR measurements.

20

3.2 Time-Resolved Two-Photon Photoemission Spectroscopy

The femtosecond TR-2PPE spectrometer has been detailed elsewhere.86 The

femtosecond laser pulses came from two home-built noncollinear optical parametric

amplifiers (NOPAs), pumped by the second and third harmonics, respectively, of a Yb-

doped fiber laser (Clark- MXR Impulse) at a repetition rate of 0.8 MHz. We carried out

the TR-2PPE measurement at room (∼300 K) and liquid nitrogen (∼80 K) temperatures

using visible pump pulses (ℏ𝜔* = 3.10 eV, 2.9 µJ cm-2) and UV probe pulses (ℏ𝜔, =

4.35 eV, 0.2 µJ cm-2) generated by frequency doubling the IR and visible NOPA

outputs, respectively. The pump and probe beams were aligned in a collinear geometry

when incident on the CsPbBr3 surface at 45° from surface normal. Using

photoemission signal at the highest electron energy (shortest lifetime), we estimated a

pump−probe cross correlation (CC) with full-width-at-half- maximum (fwhm) of ∼100

fs. On the basis of the reported absorption coefficient87 and dielectric constants,8 we

calculated an excitation density in the near surface region off 𝜌8R= 8 × 1017 cm−3 for

each excitation laser pulse. We choose this excitation density to maximize 2PPE signal,

while maintaining the linear excitation region,88 i.e., below the onsets (≥1018 cm3) of

many-body effects, such as Auger recombination, Auger heating and hot phonon

bottlenecks in LHPs.78,79,89–91 The photoelectrons were detected by a hemispherical

electron energy analyzer (VG Scienta R3000). The kinetic energy of the photoelectrons

is referenced to the instrument Fermi level, EF. We found that the alignment of the

21

CsPbBr3 conduction band to EF changed with sample temperature; this could result

from dipoles at surfaces and interfaces in the mounted sample.

3.3 Results and Discussion

In a TR-2PPE experiment, electrons are photoexcited into the conduction band at t =

0 by the visible pump pulse, ℏ𝜔* = 3.10 eV, with excess energies of ℏ𝜔, − Eg = 0.75

eV (below the second conduction band53) and subsequently ionized at a controlled time

delay (Δt) by the UV probe pulse, ħω2 = 4.35 eV. The TR-2PPE traces for CsPbBr3 at

300 and 80 K are shown in two dimensional (2D) pseudocolor (intensity) plots in

Figures 3.1, parts a and b, respectively. Also shown are vertical cuts, i.e., photoelectron

spectra at selected pump−probe delays, Figures 3.1, parts c and d for 300 and 80 K,

respectively. We make two observations: (1) The initial photoexcitation creates a broad

distribution of energetic electrons that relax within Δt ∼ 0.5 ps, and (2) there is little

change to the electron energy distribution at longer time scales and long-lived energetic

electrons, if any, are below our detection limit at Δt > 0.5 ps. On the basis of the second

observation, we take photoelectron spectra at long delay times (Δt ≥ 4 ps) as those

from thermalized electrons (with the phonon bath) in the conduction band. Note that,

following photoexcitation, electrons (and holes) equilibrate through electron−electron

scattering which takes place on the ultrashort time scale of ∼10 fs to give quasi-thermal

distributions of hot electrons characterized by an electron temperature higher than the

22

phonon temperature;76 this ultrafast process is not resolved with our experiment time

Figure 3.1. Hot electron cooling probed by TR-2PPE. (a, b) 2D pseudo color (intensity) plot

of TR-2PPE spectra as a function of electron energy and pump−probe delay for single-crystal

CsPbBr3 at (a) 300 K and (b) 80 K; (c, d) electron energy distribution curves (at representative

pump−probe delays (Δt = 0.06−2.0 ps) at 300 K (c) and 80 K (d). (e, f) Mean energy of the

electron distribution as a function of pump−probe delay at 300 K (e) and 80 K (f). Here the

mean electron energy ⟨E⟩ is referenced to thermalized electron energy, ETh, at long

pump−probe delays (Δt ≥ 4.0 ps).

1.5

1.0

E - E

F (e

V)

43210Pump-probe delay (ps)

80 K

1.00.50.01.5

1.0

0.5

E - E

F (e

V)

43210Pump-probe delay (ps)

1.00.50.0

300 K

2.01.61.20.8E - EF (eV)

0.06 0.16 0.30 0.50 1.00 2.00

ps

Inte

nsity

1.61.41.21.00.80.6E - EF (eV)

0.06 0.16 0.30 0.50 1.00 2.00

ps

0.2

0.1

0.0<E>

- µe

(eV)

2.52.01.51.00.50.0Pump-probe delay (ps)

0.2

0.1

0.0<E>

- ETh

(eV

)

2.52.01.51.00.50.0Pump-probe delay (ps)

(a) (b)

(c) (d)

(e) (f)

23

resolution.

We calculate the mean kinetic energy, ⟨E⟩, of the hot electrons directly from the

photoelectron distribution at each pump−probe delay. We reference ⟨E⟩ at short

pump−probe delays to the thermal energy, ETh which is ⟨E⟩ at Δt ≥ 4 ps. The results

(gray circles) are shown in Figures 3.1e and 3.1f for sample temperatures of 300 and 80

K, respectively. At both temperatures, the excess mean electron energy ⟨E⟩ relaxes on

ultrafast time scales of ≤0.2 ps. Single exponential fits (red curves in Figures 3.1e and

3.1f) yield time constants for electron energy relaxation of τe = 0.20 ± 0.02 ps and 0.16

± 0.01 ps at 300 and 80 K, respectively. Linear fits to ⟨E⟩ in the time window of

0.06−0.16 ps yield initial hot electron cooling rates of d⟨E⟩/dt = −0.64 ± 0.05 and

−0.82 ± 0.05 eV/ps at 300 and 80 K, respectively. Note that we include only data

points outside of the CC curve to avoid complications of photo- electron signal from

UV pump and visible probe.

The high hot electron cooling rate and its weak and inverse temperature dependence

are understood within the conventional mechanisms for hot electron cooling in

inorganic semiconductors. For excess energy substantially above the LO phonon

energy, hot electrons cool down initially in a polar semiconductor by the emission of

LO phonons, with emission rates determined by Coulomb interaction. Given the mean

excess energy (∼200 meV in Figure 3.1, parts e and f) being one order of magnitude

24

greater than the LO phonon energy (ℏ𝜔OP = 16.8 meV) in CsPbBr3 perovskite,9,92 the

−0.64−82 eV ps-1 cooling rate is similar to those in a typical polar semiconductor, such

as GaAs.

The weak and inverse temperature dependence is expected from the difference

between electronic temperature (Te) and phonon temperature (Tph), as well as the

Bose−Einstein statistics of the LO phonons. The energy relaxation rate of unscreened

hot electrons can be derived from Fermi’s golden rule, as done by Pugnet et al.70 and

more recently simplified by Fu et al.:78f

U MUV

≈ − ℏXYZ[\]YZ

^_` )ℏaYZbc\)^_` )ℏaYZbcde

*)^_` )ℏaYZbcde

ℏXYZfgh\

i9 exp ℏXYZ

,gh\𝐾E

ℏXYZ,gh\

(1)

where k is the Boltzmann constant, 𝜏8)OP is an effective electron−phonon scattering

time constant, and 𝐾E is the zeroth order Bessel function which is approximately unity

at the transient electron temperatures treated here. Our experimentally determined hot

electron cooling rates in CsPbBr3 at the two temperatures (300 and 80 K) correspond

to a common electron−phonon scattering time of 𝜏8)OP = 3.8 ± 0.4 fs at an average

excess electron energy of ∼0.15 eV.

The ultrafast hot electron cooling via LO-phonon emission should result in a

dynamic process in which hot electrons at higher energies cascades down to lower

energies. This cascading process is observed in the hot electron populations at different

25

excess electron energies, Figure 3.2, parts a and b for 300 and 80 K, respectively. Here,

Figure 3.2. Hot electron population and total photoemission intensity as a function of time.

(a, b) Hot electron populations (normalized intensities) for selected excess energies obtained

from the TR-2PPE spectra of CsPbBr3 at (a) 300 K and (b) 80 K. (c) Total photoemission

intensity as a function of pump-probe delay for 300 K (red circles) and 80 K (blue crosses).

Note that in all panels we only show data at ∆t > 0.6 ps to avoid complications due to

overlapping photoelectron signal from both visible-pump/UV-probe and UV-pump/visible-

probe when pump probe laser pulses overlap.

Norm

aliz

ed In

tens

ity E - ETh (eV) 0.6 0.5 0.4 0.3 0.2 0.1

Norm

aliz

ed In

tens

ity E - ETh (eV) 0.6 0.5 0.4 0.3 0.2 0.1

Inte

grat

ed In

tens

ity

1.51.00.5Pump-probe delay (ps)

80 K 300 K

(a) 300 K

(c)

80 K(b)

26

each curve at a particular excess energy (above ETh) is obtained from a horizontal cut

of the 2D plots in Figure 3.1, part a or b, and normalized to the energy-integrated total

TR-2PPE intensity. Since electron−hole recombination occurs on the much longer

time scale of nanoseconds,88 this normalization process removes changes in TR-2PPE

signal from variations in photoionization cross sections and/or diffusion of carriers

from the near-surface region (detectable in TR-2PPE) into the bulk of the crystal. As

expected from the hot electron cooling cascade, we observe only decrease in electron

population with time at higher electron energies. The apparent population relaxation

slows down with decreasing electron energy and, at intermediate electron energies (e.g.,

0.4 and 0.5 eV above ETh), the population rises initially followed by decay at longer

times. On the lower energy end (e.g., 0.1 eV above ETh), we only observe a rise followed

by saturation in population on this time scale. In agreement with the analysis of ⟨E⟩ in

Figure 3.1, parts e and f, the cascading processes in Figure 3.2, parts a and b, are mostly

completed for Δt ≤ 0.5 ps and are very similar at the two sample temperatures, i.e.,

weak temperature-dependence.

A major limitation of describing hot electron relaxation in LHPs by eq 1 is that

it assumes unscreened electrons.70,78 After the emission of a few LO phonons, a large

polaron can be formed to screen the Coulomb potential for subsequent scattering with

LO phonons to lose remaining excess electronic energy. This effect is particularly

important in LHPs as the real part of the dielectric constant (𝜀*) increases by one-order

27

of magnitude following the activation of optical phonons on the picosecond to

subpicosecond time scale.8 The large polaron screening has been proposed as mainly

responsible for long- lived hot carriers that retain parts of their excess energies on time

scales as long as 102 ps in hybrid organic−inorganic LHPs (CH3NH3PbI3 and

CH3NH3PbBr3).10,11 Further contribution to long lifetimes of hot carriers can come

from the phonon glass character which results in low thermal conductivity and

reheating of the electronic/LO phonon manifolds by the acoustic phonon bath.8,91,93

However, unlike hybrid organic−inorganic LHPs, long-lived hot carriers were not

observed in their all-inorganic counterpart, CsPbBr3.10 This was hypothesized as

resulting from the large polaron screening process being slower than initial hot carrier

cooling due to LO-phonon emission in CsPbBr3.9,8 Based on time-resolved optical

Kerr effect (TR-OKE) spectroscopy and transient reflectance (TR) spectroscopy, we

found that large polaron formation at room temperature occurs on the time scale of τp

= 0.6−0.7 ps,9 which is ∼3× slower than the initial hot electron cooling process (Figure

3.1e). In the following, we aim to understand the slower polaron formation in CsPbBr3

than that in its hybrid counterpart from the dependence of polaron formation rate on

temperature.

The first evidence for polaron formation comes from the dynamic change in

the nature of photoexcited electrons, as revealed by the time-dependent

photoionization cross section. In Figure 3.2c, we plot the energy integrated TR-2PPE

28

intensities as a function of pump−probe delay times at 300 K (red circles) and 80 K

(blue crosses). We see a large decrease in the total photoelectron intensity by as much

as 4× from 0.06 to 1.5 ps. Similar observations have been reported before in

photoemission studies of CH3NH3PbI3 thin-films and single-crystals.94 This change in

photoemission signal does not correspond to a loss of carrier population as

electron−hole recombination occurs on the much longer time scale of nanoseconds.

Ballistic diffusion away from the probe volume (∼nm from the topmost surface) is a

necessary consideration when one quantitatively analyzes photoemission from a single-

crystals,41 but this effect would predict faster loss of photoemission signal at lower

temperatures as expected from the inverse temperature dependence in charge carrier

mobility or diffusion constant. In contrast, the experimental results in Figure 3.2c

shows slower loss in signal at 80 K than that at 300 K. Additionally, similar decrease in

TR-2PPE intensity on sub-ps time scales is also seen from thin-film CH3NH3PbI3 of

40 nm thickness where the whole film is excited and diffusion away from the topmost

surface is negligible.11 Thus, we assign this initial loss in 2PPE signal to a dynamic

decrease in the photoionization cross section. This is consistent with the large polaron

formation process, in which the dynamic localization decreases the optical transition

strength from an electron in the conduction band to a free electron resonance above

the vacuum level. The polaron formation rate at either 300 or 80 K in Figure 3.2c is

clearly slower than the hot electron cooling rate in Figure 3.1, part e or f, although we

are not able to extract a single lifetime due to the non-exponential nature of the data in

29

Figure 3.2c. Unlike the weak and negative temperature dependence in hot electron

cooling (Figure 3.1, parts e and f), the presumed polaron formation rate is strongly and

positively dependent on sample temperature. On the basis of the initial slope (∼30%

loss in intensity from the value at 0.06 ps), we estimate the rate of change of

photoionization cross section at 80 K is ∼3 times lower than that at 300 K. Before

discussing the mechanistic origin of this temperature dependence, we corroborate this

interpretation with transient reflectance measurement at the two temperatures.

Parts a and b of Figure 3.3 show 2D pseudocolor plots of transient reflectance

spectra from CsPbBr3 single crystal at 300 and 80 K, respectively. For comparison with

the TR-2PPE data above, we use similar excitation conditions: ℏ𝜔 = 3.18 eV and

energy densities 𝜌8R = 3.0 µJ/cm2. In addition to the known red shift in the excitonic

resonance with increasing temperature,95,96 We point out the main feature in the two

spectra in Figure 3.3, parts a and b: the presence of a transient red shift in the band gap

transition, resulting in the derivative shape in the TR spectra. This transient red-shift

effect has been observed before in CH3NH3PbI3 thin films with the magnitude of red

shift known to scale with excess carrier energy;97,98 It can be attributed to band

renormalization, e.g., due to a transient Stark effect in the presence of delocalized hot

charge carriers.97 We recently showed that, for both CH3NH3PbBr3 and CsPbBr3 single

crystals, the decay of this red-shift in TR spectroscopy is concurrent with large polaron

formation observed in TR-OKE spectroscopy. Large polaron formation diminishes

30

the screening effect of free carriers on the excitonic resonance, leading the time-decay

in band renormalization.

We now turn to the temperature dependence of the polaron formation

Figure 3.3. Transient reflectance spectroscopy revealing polaron formation. (a-b)

Pseudocolor 2D plots of transient reflectance spectra of a CsPbBr3 single crystal pumped by

ℏ𝜔 = 3.18 eV and a pulse energy density of 𝜌8R = 3 µJ cm-2 at sample temperatures of (a) 300

K and (b) 80 K. (c) Kinetics of transient red shifts extracted from vertical cuts of the

pseudocolor 2D plots at 528 nm for 300 K (red) and 518 nm (80 K). The open circles are

from measurements at𝜌8R = 3 µJ cm-2 and the solid circles are from a lower excitation density

of 𝜌8R = 0.8 µJ cm-2. The gray area shows the pump-probe cross correlation. The solid curves

(red, 300 K; blue, 80 K) are single exponential fits (convolved with the CC) that give the decay

time constants of 𝜏5 = 0.7 ± 0.1 and 2.1 ± 0.2 ps at 300 K and 80 K, respectively.

31

dynamics in CsPbBr3 single crystals from the transient red-shifts. A comparison of TR

spectra at 80 and 300 K in Figure 3.3a reveals that the transient red-shift at 80 K is

much longer lived that that at 300 K. This is quantified in vertical cuts at the negative

peaks in the 2D plots. Note that for 𝜌8R ≥ 3.0 µJ cm-2, hot carrier cooling dynamics is

slowed down due to many-body effects of Auger heating and phonon bottleneck

effects.79,89,90 In Figure 3.3c, we compare the dynamic red-shifts at two excitation

densities: 𝜌8R = 0.8 µJ cm-2 (solid circles) and 𝜌8R = 3.0 µJ cm-2 (open circles) for each

temperature. The high excitation density of 𝜌8R = 3.0 µJ cm-2 is at the threshold for

many-body efforts, as reflected in the slight slowing-down in the time profiles, as

compared to those at 𝜌8R = 0.8 µJ cm-2. We analyze the time profiles at 𝜌8R = 0.8 µJ

cm-2 to obtain the polaron formation time constants. At both temperatures, the time-

dependent red-shifts can be well described with single exponential decays (solid curves

in Figure 3.3c) with time constants of 𝜏5 = 0.7 ± 0.1 and 2.1 ± 0.2 ps at 300 and 80 K,

respectively. While the 𝜏5 value at 300 K is in excellent agreement with previous TR-

OKE and TR measurements,9 the temperature dependent rates in Figure 3.3c are also

consistent with the results in Figure 3.2c on the dynamic change in photoionization

cross sections. Both measurements reveal that large polaron formation rate increases

with temperature, with 𝜏𝑝 decreasing by a factor of 3× from 80 to 300 K.

Since CsPbBr3 is in the orthorhombic phase at both 80 and 300 K,99 the

different polaron formation dynamics at the two temperatures does not have a

32

structural origin. We can understand the temperature dependent polaron formation

rates in CsPbBr3 by considering the effects of phonon disorder.60,45 Here, we follow

the argument of Ku et al. in their quantum dynamics treatment of a small polaron.84

We consider a bare electron of momentum k and its overlap with large polaron states

to give the polaron spectral function. For a large polaron state with momentum k, there

are a continuum of polaron states of momentum 𝑘 − 𝑞 along with the absorption or

emission of phonons with moment 𝒒. For simplicity, we consider the weak coupling

case where multi-phonon transitions are ignored. In this limit, the polaron spectral

function will contain a line shape similar to the phonon line shape. When the

propagator in the polaron basis is applied to the bare electron wave function, one can

in principle obtain the temporal decay from the initial bare electron state. As an

approximation, we estimate this decay time constant, the polaron formation time, from

the inverse line width based on the uncertainty principle. Indeed, low-frequency Raman

measurements on CsPbBr3 single crystals revealed the key characteristic of dynamic

disorder in phonon modes and this dynamic disorder increases with temperature. The

line width of the PbBr3− phonon modes (responsible for polaron formation9) in

CsPbBr3 increase from ∼5 to ∼12−15 cm−1 as the temperature is increased from 80 to

300 K.60 This increase in phonon line width is in almost quantitative agreement with

our measured decrease in large polaron formation time constants at the two

temperatures.

33

The results presented here also explain why long-lived hot carriers are observed

in hybrid LHPs but not in their all- inorganic counterparts.10,11 In CsPbBr3, large

polaron formation is 3× slower than initial cooling of bare hot electrons/ holes by LO-

phonon emission; thus large polaron screening is not dynamically competitive in

sufficiently slowing down the cooling process. In hybrid LHPs, large polaron formation

occurs on similar time scales (𝜏𝑝 = 0.28 ± 0.04 ps in CH3NH3PbBr3)9 as that of the

initial cooling of bare hot electrons/holes11 by LO-phonon emission. Based on the

estimated large polaron binding energy,9 one can predict that emission of only a few

LO-phonons is sufficient to lead to the formation of a large polaron. The resulting one

order-of- magnitude screening of the Coulomb potential drastically slows down further

LO-phonon emission. Loss in the residual excess electron energy in the hot large

polaron is further slowed down due to low thermal conductivity of hybrid LHPs; in

other words, the local hot acoustic phonon bath can reheat the electron/hole and LO-

phonon manifolds.91,93 While Frost et al.93 predicted a higher thermal conductivity in

all-inorganic LHPs than their hybrid counterparts, experimental measurements by

Elbaz et al.100 revealed the same low thermal conductivity in both CH3NH3PbBr3 and

CsPbBr3. Thus, the competitiveness in the initial hot carrier cooling and large polaron

formation dynamics is likely the main reason for long- lived hot carriers. The above

discussions pertain to low excitation densities. At high excitation densities (≥1018 cm-

3), many body effects may become dominant and hot carrier cooling slows down with

34

increasing excitation density.79,89,90

35

Exciton-Polaritons and Continuous-Wave Lasing 4

An ideal material for a solar cell approaching the theoretical Shockley-Queisser limit must also

be an excellent light emitter with near unity quantum efficiency.101 Lead halide perovskites, a

class of materials that have aroused much imagination in recent optoelectronics research, fulfill

this requirement.102 Indeed, lead halide perovskites have been used not only in solar cells2 with

performance approaching those of Si and GaAs, but also in light-emitting diodes and lasers

with high efficiencies.3,4 We focus on a particular class of optoelectronic devices, nanowire

(NW) lasers, that may serve as sub-wavelength coherent light sources for a wide range of

applications.103,104 Lead halide perovskite NW lasers33–37 are particularly attractive because (i)

they can be easily grown from solution at room temperature; (ii) they are broadly tunable in

color by mixing or exchange of cations or anions, covering the complete visible and near-IR

wavelength region; (iii) they show lasing at carrier densities as low as 1016 cm−3, which is two

orders of magnitude lower than those in conventional semiconductor NWs.33 However, all

lead halide perovskite NW lasers demonstrated thus far are limited to pulsed operation. While

continuous wave (CW) lasing is vital to achieving electrically injected NW lasers and to

applications such as optical communication and spectroscopy,105,106 Achieving CW lasing in

lead halide perovskite NWs was thought to be exceptionally challenging due to thermal

damage and to screening of the exciton resonance.107,108 The latter leads to a reduction in

oscillator strength and gain at high excitation densities necessary for population inversion.107,108

36

However, lasing in a NW cavity may not necessarily require population inversion.

Coupling between the exciton resonance and light is enhanced in a NW due to the reduced

mode volume of the photons and the cavity enhanced oscillator strength through the relation

Ω ∝ f/V , where Ω is the vacuum Rabi splitting; f is the oscillator strength; and V is the mode

volume.109 In this regime, coherent light emission can originate from the steady-state leakage

of an exciton–polariton condensate below the threshold for population inversion.110 An

exciton–polariton, or polariton for short, is a bosonic quasiparticle formed by the

superposition of strongly coupled exciton and photon states, which generates an upper and

lower polariton branch (UPB and LPB, respectively) from the avoided crossing of the two

dispersions.111,112 The LPB is exciton like at high momentum (k) and photon like at lower k

(see Figure 4.1). Consequentially, polaritons relax along the LPB by acoustic phonon emission

and accumulate near the avoided crossing, i.e., the polariton bottle- neck, due to the reduced

lifetimes and density-of-states in the photon-like region at lower energies.112 Polaritons

undergo Bose-stimulated scattering, which surpasses spontaneous- scattering at a critical

density to produce the coherent condensate state113 and the light leaking out of the cavity from

such a coherent state has been called polariton lasing.111,112 Condensation in the bottleneck

region was observed in CdTe microcavities as a ring of emission in angle-resolved fluorescence

measurements.114,115 The NW cavity differs from the planar microcavity. While both are Fabry–

Perot cavities defined by end facets in a NW37,103,104 or distributed Bragg reflectors (DBRs) in

a conventional microcavity,73,111–113 the former features sub-wavelength lateral dimensions

(≈102 nm),103,104 as opposed to the µm to mm dimensions of the latter. Light escaping the NW

37

cavity at the end facet is diffracted, therefore making traditional angle- resolved measurements

infeasible.116 In addition, the detuning between exciton and cavity photon in a NW differs

from the near resonant conditions commonly engineered in DBR microcavities. Resonant

Figure 4.1: (a) Dispersions of the cavity photon (blue) and exciton (green) of the perovskite

NW without coupling. The black and grey curves are the LPB and UPB with W = 200 meV.

The red dots show the experimental polariton modes near the bottleneck region of the LPB.

The blue dashed line is the photon dispersion outside of the NW, illustrating the momentum

mismatch between a free photon and a waveguided polariton; (b) relative populations of the

phonon (red) and the exciton in the polariton state; (c) density of states (DOS) in one

dimension are plotted for the cavity photon (blue) and polariton (red) dispersions. (d) Mode

spacing for photon (blue) and polariton (red) dispersions. The curves are proportional to

the first derivatives of the respective dispersions.

38

conditions in a NW would require a larger degree of cross-sectional confinement and would

be impractical due to the loss of waveguiding at such dimensions. Negatively detuned

waveguide polaritons have been detected in both inorganic109,117 and organic NWs.118

The continuing search for materials with strong light-matter interactions119,120 has

recently turned to lead halide perovskites. Park et al. reported strong light-matter interaction

in CsPbBr3 perovskites based on the uneven mode spacing in pulsed lasing from these NWs.37

Su et al. fabricated CsPbCl3 planar DBR microcavities near resonant conditions and the

measured polariton dispersions gave a large Rabi splitting of 2Ω = 265 meV.73 More

importantly, these authors demonstrated unambiguously polariton lasing at room temperature

under pulsed conditions at high excitation densities.73 These demonstrations have motivated

us to explore CW lasing in lead halide perovskite. Here we demonstrate CW lasing from

cesium lead bromide perovskite (CsPbBr3) NWs at excitation densities as low as 6 kW cm-2

and with effective quality factors as high as 103. We find that the emission spectrum from each

NW is characterized by a non-linear progression of sub-bandgap modes,117 with energy spacing

more than an order of magnitude smaller than those of cavity photon modes and with a

negative curvature in the mode dispersion U9MUg9

< 0 , as predicted for the bottleneck region

on the LPB (see Figure 4.1). In our NW geometry, the cavity photon is negatively detuned and

located at 1.2 eV below the exciton resonance; thus, polariton condensation is expected to

form in the bottleneck region, but not at k ≈ 0. With increasing excitation power, we observe

a distinct threshold, above which the intensity of a few polariton modes in the bottleneck

region grows superlinearly, consistent with lasing. Analysis of the nonlinear polariton

dispersions gives vacuum Rabi splitting of Ω = 0.20 ± 0.01 eV, consistent with strong light–

39

matter interaction in both 3D37,73 and 2D layered lead halide perovskite.121–123 These findings

represent another demonstration of the superior optoelectronic properties of lead halide

perovskites.

4.3 Results and Discussion

We studied single-crystal CsPbBr3 perovskite NWs and nanoplates (NPs) grown from

solution, as described elsewhere34 see Figure 4.2, for detailed characterization). The NWs and

NPs were dispersed onto a sapphire substrate from the growth surface by physical contact.

We focused on NWs that were mostly of rectangular shape, with subwavelength cross-

Figure 4.2: (a) (b) (c) AFM images of individual perovskite nanowires (NWs) with different

length, (a) 5.0 µm (b) 8.5 µm (c) 14.0 µm; (d) (e) (f) Height profiles across the top and bottom

of the NWs in (a) (b) (c) respectively, indicating the width and height of each NWs.

40

sectional dimensions of 200–500 nm and length, L = 4–20 µm; see Figure 4.2 for atomic force

microscope (AFM) images. We carried our optical measurement in a cryostat under vacuum

(10−6 torr) using a home-built far-field epi fluorescence microscope with the sample stage at

77 K. A CW laser diode with 1 W output at 450 nm (ℏ𝜔 = 2.76 eV) was used to uniformly

illuminate individual NWs while fluorescence spectra and images where collected.

Figure 4.3a shows fluorescence spectra from a 20 µm long CsPbBr3 NW at CW

excitation power densities of P = 0.25–7.8 kW cm−2. As P increases, a peak on the low energy

side of the fluorescence spectrum grows in and, for P ≥ 3.5 kW cm−2, a progression of small

sub-bandgap peaks, reminiscent of Fabry–Perot cavity modes but with one order of magnitude

smaller spacing (see Figure 4.1). As we show below, the positions of these peaks are

determined by a nonlinear dispersion with negative curvature characteristic of polaritons. This

series of small peaks continues to grow in intensity until a threshold excitation power of Pth ≈

6 kW cm−2, above which 2–3 modes, retaining their mode spacing, become dominant and

grow at a much faster rate than the rest of the spectrum. This is more obvious when we plot

the fluorescence intensity integrated in an energy window including the dominant modes

(2.32–2.33 eV), Figure 1b: the slope above 6 kW cm−2 is nine times that below this threshold,

consistent with polariton lasing. The full-width-at-half-maximum of the dominant lasing peak

at 2.3238 ± 0.0002 eV is 0.0010 ± 0.0001 eV, giving an effective quality factor of Qeff = 2300

± 300, similar to that of pulsed lasing in the same NWs.34 Note that the exact number of modes

that are enhanced above the polariton lasing threshold varies from NW to NW (see Figure

4.1), as does the polariton lasing thresholds, which is likely due to differences in the details of

cavity geometry.

41

Coherent light emission is evident from the interference pattern in fluorescence

images. The images on the right side of Figure 1 show three NWs (L = 20, 14, and 5 µm)

above their respective lasing thresholds. Such interference pat- terns are not observed below

the thresholds. As explained by Van Vugt et al. for similar images from pulsed lasing in ZnO

NWs, each pattern results from the interference of diffracted coherent light escaping from

Figure 4.3: (a) PL spectra of a 20 µm long NW obtained with increasing excitation light

power densities in the range of 0.25-7.8 kW cm-2. Inset (b) shows the integrated power

density plotted against the power density, showing the onset of nonlinear growth in the

modes near ~ 6 kw/cm2. The right side shows fluorescence images above the polariton

lasing threshold for NWs with length (c) 20 µm, (d) 14 µm, and (e) 5 µm. The scale bad in

(c) is 5 µm.

42

two end facets of the NW cavity.116 We carry out a simple simulation using two coherent point

sources separated by L, and the resulting pat- terns reproduce key features, namely the spatial

distribution and fringe spacing away from the NW.

We confirm the progression of peaks in the fluorescence spectra originate from cavity

polariton modes. Figure 4.4a–c compares fluorescence spectra near lasing thresholds from

three different NWs with L = (a) 20, (b) 14, and (c) 7 µm, respectively. While the mode spacing

(Δ) in general increases with decreasing cavity length, the results do not agree with the photon

modes of a subwavelength Fabry–Perot cavity where Δ is a constant at each L. In contrast,

the observed spacing decreases with increasing mode index, j (Figure 4.4d), as expected from

the negative curvature of the LPB dispersion in the bottleneck region (see Figure 4.1d); the

polariton model, with negative curvature in the dispersion in the bottleneck region, will be

detailed later (Figure 4.6).

The nonlinear dispersion may be given a dual interpretation: a classical one in which

the propagation velocity of a traveling light wave is reduced due to an enhanced refractive

index as the frequency approaches that of the exciton resonance and a quantum one described

above in the polariton model. Though these are equivalent models, the quantum polariton

model allows us to extract an interaction strength directly and is used here. We quantitatively

analyze the exciton–polariton dispersion in CsPbBr3 NWs using the Hamiltonian for exciton–

photon coupling in the LPB111,112

𝐸) =12 𝐸8R + 𝐸5s −

12 𝐸8R − 𝐸5s

, + 𝛺,(4.1)

43

where Ω is the vacuum Rabi splitting; Eex is exciton dispersion, approximated as a constant

due to its much heavier mass compared to that of the photon; and

𝐸5s =𝑐𝑛 ℏ,𝑘, + 𝑚E

, 𝑐𝑛

,(4.2)

Figure 4.4: Panels (a) - (c) display fluorescence spectra for NWs of length L = (a) 20 µm,

(b) 14 µm, and (c) 7 µm, respectively. Lines are shown to mark the positions of individual

peaks. The difference between peak positions is plotted against the reciprocal nanowire

length in panel (d).

44

is the cavity photon dispersion; 𝑐 is the speed of light; 𝑛 (=2.3) is the refractive index36; ℏ is

Planck’s constant; 𝑚0 is the cavity photon mass determined by the confinement imposed by

the subwavelength cross section where 𝑎 (=330 ± 100 nm) is the average cross-sectional

dimension of the NWs determined from AFM images of 12 representative NWs (Figure 4.2).

While the uncertainty in 𝑎 can change the zero-point energy of the cavity photon by ±30%,

its effect on the bottleneck region is much smaller. Because of the large detuning energy of

the system, photon dispersion near the bottleneck region is almost linear; this, along with the

constant exciton energy in the 𝑘∥ range of interest, means that a change in 𝑚0 (Equation (3))

corresponding mainly to an offset in the 𝑘∥ value in the crossing region. We estimate that the

uncertainty in zero-point photon mass introduces ±5% in the value of Rabi splitting obtained

from fitting to the experimental data as detailed below.

We determine the peak positions of individual polariton modes by fitting the PL

spectra with a sum of Lorentzian functions, in addition to the broad fluorescence background.

We show in Figure 4.5 fluorescence spectra (red), along with the total fits (blue) and individual

modes (black) above the polariton lasing thresholds for two representative NWs of L = (a) 14

µm and (b) 11 µm. The resulting peak positions, i.e., mode energies, Figure 3c, show the

distinct negative curvatures characteristic of the bottleneck region on the LPB. We t the data

in Figure 3c to the exciton–polariton dispersion in Equation (1), where k|| = πj/L + k0 and j

(= 0,1, 2, ...) is the mode index from the experimental data. We treat k0 as a fitting parameter

to account for the absence of knowledge on the wavevector of the lowest energy mode and

the uncertainty in zero-point photon mass also introduces mainly an uncertainty in the offset

in k. The fits yield vacuum Rabi splitting of Ω = 196 ± 2 and Ω = 198 ± 2 meV for L = 14

45

and 11 µm, respectively. We also carried out analysis below and above the polariton lasing

thresholds and found that, within experimental uncertainty, the Rabi splitting is relatively

Figure 4.5: Representative PL spectrum (red) above the lasing threshold for NWs of (a) L

= 14 µm and (b) L = 11 µm, at excitation power densities of 9.9 and 19.5 kW cm-2,

respectively. The fits (blue) and deconvoluted Lorentzian peaks (black) are also shown. (c)

Peak positions (circles) obtained from PL spectra with the polariton model fits (solid curves)

for L = 14 µm (blue) and 11 µm (red) NWs, respectively. (d) Histogram of Rabi splitting

from 21 dispersions (15 different nanowires).

46

insensitive to NW length or excitation density. Results from the analysis of 21 dispersion

curves give vacuum Rabi splitting Ω in the range of 175 to 235 meV, with a mean value of Ω

= 200 ± 12 meV, as shown by the histogram in Figure 3d. This uncertainty is within the ±5%

estimated error range for the Rabi splitting, resulting from uncertainty in zero-point photon

mass. Note that, while we also let Eex be a fitting parameter, the results of the fitting give Eex

= 2.361 ± 0.005 eV, in excellent agreement with the known excitonic resonance of CsPbBr3

in absorption spectra.95,96 There is little change to the fitting results when we keep Eex = 2.361

eV constant.

Within the limited power range investigated, there is a blue-shift in polariton modes

with excitation density. This is similar to what has been observed in other polariton systems

and attributed to repulsive interaction between exciton–polaritons.124,125 It is also important to

note that the blue shift persists across the lasing threshold, with no measurable change in mode

spacing. Thus, the same polariton dispersion accounts for the observed cavity modes both

below and above the lasing threshold.

The complete polariton model is shown in Figure 4.6 with the experimentally observed

bottleneck polariton modes displayed as red dots. Here the exciton energy at 77 K was

obtained from ref. [38] and confirmed in the absorption spectrum from CsPbBr3 single

crystals. Note the LPB in the photon-like lower k (≤2 × 10−3 A−1) region is fixed by the

physical geometry of the NW and the higher k region (>3 × 10−3 A−1) by the exciton energy.

Only the bottleneck region is determined from fitting to experimental data. This model is

consistent with the kinetic condensation regime for negatively detuned cavities.113

47

Room temperature CW polariton lasing is highly desirable from an application

perspective, but we find that the NW samples quickly degrade at temperatures above 77 K

under CW excitation. The average heat load and, thus, photodegradation, can be much

reduced if the NWs are excited under pulsed conditions and this has indeed been

demonstrated in room temperature lasing for lead halide perovskite NWs33–37 or nanoplates in

DBR microcavities.73 To carry out quantitative analysis over a broad temperature window, we

return to pulsed lasing in CsPbBr3 NWs. The pulsed excitation is at 454 nm, with a pulse width

of 60 fs, and rep-rate of 0.5 MHz. Figure 4.7 shows PL spectra near and above the lasing

threshold under pulsed excitation at three representative temperatures: (a) 77 K, (b) 171 K,

and (c) 295 K. At all temperatures, we observed lasing behavior as represented by the nonlinear

growth in emission intensity of a few sharp peaks, similar to the CW lasing in Figure 1. The

most striking observation is the increase in mode spacing by approximately one order of

magnitude when the temperature is increased from 77 to 295 K. Thermal expansion is not

responsible for this change in mode spacing with temperature. In a Fabry–Perot cavity, the

mode spacing ΔE given by fℏ{7O

, where h is the Planck constant, c is the speed of light, n is the

group index, and L is the length of the NW (cavity). Based on the measured thermal expansion

coefficient of 3.8 × 10−5 K−1 for CsPbBr3,126 we calculate a mode spacing change of −0.8% as

the temperature is increased from 77 to 295 K; this contrasts the +500% change observed in

experiments. In addition, the main lasing modes do not follow the temperature dependence

of the band gap but rather red-shift with increasing temperature. Figure 4.7d shows the

48

experimental dispersions (circles) at three temperatures. Due to a small blue-shift in exciton

resonance from 2.360 eV at 77 K to 2.396 eV at 295 K, we plot the dispersions referenced to

Eex at each temperature. Using the same analysis as in Figure 3, we t the modes at each

temperature here to the LPB in Equation (1) and the results for the same NW at three different

temperatures yield Ω = 200 ± 10 meV, which is identical to that obtained for CW lasing. We

emphasize that the experimental results of lasing modes at all temperatures are well described

by a single dispersion curve of the LPB. At higher temperature, the more efficient cooling of

Figure 4.6: Dispersions of the cavity photon (blue dashed) and exciton (green) at 77 K of

the 14 µm long NW without coupling. The black curves are the lower (LPB, dashed) and

upper (UPB, solid) polariton branches formed with W = 200 meV. The red dots show the

experimental CW polariton modes near the bottleneck region of the LPB. The blue solid

line is the photon dispersion outside of the crystal illustrating the momentum mismatch

between states leading to efficient waveguiding.

49

the polaritons by acoustic phonon scattering moves the condensate state further down the

bottleneck region on the LPB, i.e., closer to the photon-like region. This explains both the

decrease in energy and the increase in mode spacing with temperature.

4.2 Conclusion

The observation of CW lasing from polariton modes in CsPbBr3 NWs reveals a strong light–

matter interaction. We made similar observations in NPs with subwavelength thicknesses of

200–500 nm and lateral dimensions of 2–20 µm. The Rabi splitting of Ω = 200 meV

determined here for CsPbBr3 NWs is very similar to values from 2D layered organic–inorganic

hybrid perovskite121–123 or 3D CsPbBr3 nanoplates73 in DBR microcavities, suggesting that

strong light–matter interaction is intrinsic to this class of materials. The exciton binding

energies in the 2D layered hybrid perovskites are higher than those in their 3D counterparts;

we may expect stronger light–matter interaction in the former and potentially lower exciton–

polariton lasing thresholds. The vacuum Rabi split- ting measured for CsPbBr3 NWs

represents the highest value for inorganic semiconductor lasers where previous studies

reported Ω up to ≈25 meV in the near-IR to visible region in, e.g., GaAs and CdTe, and up

to ≈150 meV in the UV region for large bandgap materials, such as ZnO and GaN.120 A large

Rabi splitting indicates a large oscillator strength, which is positively related to exciton binding

energy. In perovskites, while an above gap excitation creates predominantly charge carriers at

room temperature,16,88 there is a strong excitonic resonance in absorption and fluorescence

emission, with an exciton binding energy of ≈40 meV for CsPbBr3 perovskite.95,126 Thus there

50

is an equilibrium between free carriers and excitons, with the latter more favored at lower

temperatures.16 In addition to a large Rabi splitting, maintaining a polariton condensate also

requires a sufficiently low dephasing rate. In this regard, lead halide perovskites possess a

Figure 5. PL spectra under pulses laser excitation (454 nm, ~60 fs, 0.5 MHz) of an L = 13

µm CsPbBr3 NW at: (a) 77 K with 2.3, 4.7 and 7.0 µJ cm-2; (b) 171 K with 18, 30, and 42 µJ

cm-2; (c) 295 K with 40, 79, and 99 µJ cm-2. (d) Dispersions above the lasing thresholds at

temperatures ranging from 77 K to 295 K. Colored circles are experimental data points (peak

positions) and dash-black curve is the expected LPB for Ω = 200 meV. Also shown are

dispersions of the cavity photon (blue dashed), exciton (green) of the perovskite NW without

coupling. The solid grey curve is the UPB.

51

distinct advantage as recent experiments revealed the strong screening of the Coulomb

potential in this class of dynamically disordered material.10,127 Strong coupling prevails when

the vacuum Rabi frequency (Ω/ħ) is larger than the exciton dephasing rate (Γ ), Ω > ħΓ.

Previous studies on polariton lasing have shown the presence of two threshold

behavior in some systems: the first threshold at lower excitation density for polariton lasing

and the second threshold at higher excitation density for photon lasing from population

inversion.111,120 In our CPbBr3 NWs, we were not able to reach the second threshold and

observe the conversion of polariton lasing to photon lasing due to catastrophic damage to

NWs. Improve cooling of individual NWs is needed to observe the second photon lasing

threshold at high excitation densities.

In view of the strong light–matter interaction, it is possible that polaritons exist at

room temperature in bulk lead halide perovskites. It has been shown in CdS that the vacuum

Rabi splitting is reduced by a factor of ≈2.3 in going from a NW to bulk.109 This would

correspond to a bulk splitting of 87 meV with further reduction due to the increased exciton

dephasing rate at room temperature. If radiative recombination in perovskite produces

polaritons, it is intriguing to speculate that the light effective mass and sub-bandgap energies

may contribute to the observed photon recycling,128 which is beneficial to solar energy

conversion.102 These effects would be more prominent in thin-film waveguides. Additionally,

it has been shown that the exciton conductance is increased by orders of magnitude when

strongly coupled to light,129 which could contribute to carrier diffusion lengths. Thus, the

observation of CW polariton lasing with exceptionally large vacuum Rabi splitting opens the

door not only to low power CW coherent light sources and model systems for polaritonics,

52

but also to a better understanding of this class of material for solar cells.

53

Concluding Remarks 5

The prospect of building semiconductors bottom-up is a fascinating one that would allow

chemists to control parameters such as local vibrations and electronic energies of chemically

defined units that can be assembled to produce favorable properties for novel devices.130–133

To achieve this we will need to know a great deal about how the strengths and response times

of certain interactions of specific modes can lead to a large polaron, for example.

Honing in on the role of the soft lattice in large polaron formation times, TR-2PPE is

used to probe the electron distribution in single crystal CsPbBr3, showing that electron

relaxation occurs before large polaron formation in this system accounting for the absence of

hot carriers. Using TA, the transient bandgap renormalization induced by carrier screening is

probed to infer large polaron formation times showing that large polaron formation is assisted

by the dynamic disorder at higher temperatures resulting in faster formation. One would need

to increase these fluctuations to rapidly form hot long-lived polaron distributions for hot

carrier applications.

The relaxation dynamics in LHP nanowires remains highly ambiguous and is my guess

that they are drastically different from bulk LHPs since strong coupling to optical modes can

dramatically change excited state energies and lifetimes. Though there are a lot of exciting

tests, such as energy funneling and super-fluidity, left for the nanowire systems and it remains

to be seen what role polaritons could play in bulk LHPs, specifically with respect to the photon

reabsorption processes in waveguiding films that can improve PV efficiency.

54

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