= ( + )/(√

[ +

])

.

.

= /

. .

+

. .

Figure 2.1: Cubic perovskite structure of ABX3

= . = .. . .. . .. . .

Table 2.1: Estimation of A cation radius in APbX3.28,29 Radius of Pb2+ is . .34

= .

= .

≈

Inte

nsity

(a.

u.)

2422201816141210

2theta (degrees)

110In

tens

ity (

a.u.

)

4540353025201510

2theta (degrees)

100

200

Figure 2.2: X-ray diffraction patterns for MAPbI3 and MAPbBr3 powders.

. %

. %

%

%

%

. %

%

% . %

=

( ( )−

)

Figure 2.3: Photoinduced charge carrier generation and collection in a spherical perovskiteparticle infiltrated in the mesoporous scaffold. is the diffusion length of an exciton or acharge carrier. Electrons are captured by injection in the conduction band of the oxide. ,pore radius; , reactive boundary; , radius of reactive boundary.

%

%

%

−

◦

. %

Figure 2.4: (a) Mesoscopic perovskite solar cell with a mesoporous TiO2 layer and (b) pla-nar structure. In the mesoscopic device, electrons can be collected both directly throughthe TiO2 layer and through the perovskite.

25

20

15

10

5

0

Cur

rent

den

sity

(mA

/cm

2 )

1.00.80.60.40.20.0

Voltage (V)

Forward Reverse

Figure 2.5: Hysteresis of the current-voltage characteristics of a perovskite solar cell.

. %

. %

. %

%

/✷

. % .

. %

. % %

% . %

. % .

. .

.

%

. .

. .

∗ = ∗ ∗/( ∗+ ∗)

− −

. %

%

%

− / /

−

. × − −

−

. .

.

− . − .

. .

( ) = ( / )

Bandgaptuneability

WAVELENGTH (nm)1100 800 600 500 400

MASnI3MAPb1-xSnxI3

FAPbI3MAPbI3

MAPbI3-xBrx

MAPbBr3

MAPbBr3-xClxMAPbCl3

1 1.5 2 2.5 3ENERGY (eV)

Figure 2.6: Band gap of halide perovskites in the visible region and near infrared.

. .

( )

(! ) ∝√

(∑

+

)

∝√

[∑ (

! −)

+

∫ ∞ (! −

)+ ( − )

−−

!

−

⎤

⎥⎦

= −( / ) ( / )

( − ) = ( /! ) +(( /! )

)

( ) = (! / )

=( )

∫( )

ABSO

RBAN

CE0.40.20.0

E-EX (eV)

EB = 0 20 meV 40 meV 60 meV 80 meV

INTEGRATED ABSO

RPTION

80604020EB, Γ (meV)

variable EB, constant Γ variable Γ, constant EB

ABSO

RBAN

CE

0.40.20.0E-EX (eV)

Γ= 15 meV 30 meV 45 meV 60 meV 75 meV

a) b) c)

Figure 2.7: UV-vis absorption spectra computed according to the Elliott formula with (a)the line width , and (b) the exciton binding energy as a variable. Numerical values forthe parameters in the computation were chosen to represent those typical of MAPbBr ,that is = meV in panel (a), = meV in panel (b), and = . eV in all threeof the panels. Spectra are plotted versus the energy difference with respect to the =exciton transition and normalized at their value for = + . eV. Panel (c) shows thatthe integral of the normalized absorption spectra = ( / ( ))

∫( ) grows as a

function of the exciton binding energy (red line), while it is constant when only the linewidth varies (blue line).

≪

/(! ) /

(! ) /

(! ) ∝(! ) / )

ABSO

RBAN

CE

0.20.10.0-0.1-0.2E-EX (eV)

80 K 100 K 120 K 140 K 180 K 200 K 220 K 240 K 260 K 280 K 300 K

0.2

0.1

INT.

ABS

. (eV

)

-0.10

-0.05

0.00

0.05

PEAK

SHI

FT (e

V)

300200100TEMPERATURE (K)

ortho tetragonal

Figure 2.8: Analysis of the experimental absorption spectra for MAPbI films. (a) UV−visabsorption spectra for different temperatures, normalized at the energy = + .eV. The gray dotted line is the continuum contribution calculated according to Elliotformula with ( /! ) = . eV− and = . Spectra are shown with 20 K tempera-ture interval for clarity purposes, althoughmeasurements were taken at 10 K intervals. (b)Normalized absorbance integrated up to energy (defined as = ( / ( ))

∫( )

in the text) as a function of temperature. (c) Shift of the excitonic peak in the absorptionspectra as a function of temperature; when the exciton peak was not well-resolved in ab-sorption, the energy of the photoluminescence peak was taken instead, taking advantageof the negligible Stokes shift. The vertical dashed line marks the temperature of the phasetransition.

= ( ±= ( ±

= ( ±

80

70

60

50

40

30

20

10

0

EXCI

TON

BIND

ING

ENE

RGY

(meV

)

300250200150100TEMPERATURE (K)

MAPbBr3: EB ΓMAPbI3: EB Γ

Figure 2.9: Exciton binding energy versus temperature in MAPbI (filled red circles) andMAPbBr (filled green circles) films. The error bars are given on the first point for eachmaterial. Empty circles are the line width as extracted from the absorption spectra withthe formula =

∫( ) )/( ( ).

,

,=

(

!

) /− =

= .

−

−

= − − −

PL IN

TEN

SITY

(arb

.uni

ts)

43210

TIME (ns)

a)

log(

PL)

151050 TIME (ns)

PL0 (

arb.

uni

ts)

81

2 4 6 810

2 4 6

LASER PULSE FLUENCE (µJ/cm2)

b)

Figure2.10: Time-resolved photoluminescence fromaMAPb thin film. (a) Transient pho-toluminescence signal at various injected electron−hole pair densitis at the film surface.Straight lines in the semilogarithmic plot represent an exponential fit to the initial decay ofthe photoluminescence signal. Photoluminescence was excited by 150 fs long laser pulseswith a repetition rate of 1 kHz and 3.18 eV photon energy. Injected carrier densities, fromtop to bottom: . · , . · , . · , and . · cm− . (b) Photoluminescenceemission intensity estimated at time = after excitation (PL ) as a function of laserpulse fluence. The quadratic dependence is shown by the line as a guide for the eye.

( )

( ) ∝ =

≈ ≫

−

Figure 2.11: CIE diagram showing the coordinates of CsPbX3 NCs. Reprinted with permis-sion from [40].

. %

. :

%

%

. .

, − . %

. %

%

%

. %

. %

. %

.

. %−

Wavelength (nm)1100 800 600 500 400

1 1.5 2 2.5 3Energy (eV)

Nd:YAG/Nd:YLF/Nd:YVO

Alexandrite

Ti:sapphire

Dye

Nd:glass/Nd:YAG/Yb:YAG

GaAs/AlGaAs

Color Center

InGaAsP

Ti:sapphire

GaN

Figure 2.12:

− −

−

Table 2.2: Comparison of optoelectronic properties of optically active me-dia.10, 136, 226, 227, 256, 293, 296--298

−

−

0.1

1

Qua

ntum

yie

ld

1016

1017

1018

1019

1020

Injected carrier density (cm-3)

10-1

100

101

102

103

Laser pulse fluence (µJ/cm2)

Trapping Auger

Figure 2.13: Long

( )

. × −

. %

%

×. × −

. ×

.

− −

−

−

−

−

Figure 3.1: a) Sketch of the Rashba effect on optical transitions. The conduction band issplit into two, while the splitting of the valence band is assumed to be negligible, so thatthe lowest-energy optical transition is indirect; direct transitions at higher energy are stillpossible. Bands with the same colour have the same spin polarization. b) Absorptancespectra for MAPbI3 and MAPbBr3 thin films (red and green dashed lines, respectively), to-getherwith thephotoluminescence spectrumcalculated as described in themain text fromthe absorptance using reciprocity relations (red and green continuous lines), and the effec-tive photoluminescence measured from samples (black lines).

Figure 3.2: Sketch of the measurements of radiative recombination rates. A femtosecondoptical pulse excites electrons in the conduction band and holes in the valence band thanksto interband absorption. After ultrafast relaxation to the band edge, the densities of pho-toexcitations are equal in the two bands and can be calculated as the product of the opticalabsorption coefficient times the photon flux. The intensity of the prompt optical emission

generated by the photoexcitations is unaffected by traps or other non-radiative re-combination channels and can be written as the product of the bimolecular radiative re-combination coefficient times the concentration of electrons and holes.

(! )

(! ) ∝ (! )! (−! / )

=

= ,

=

(! )

= (! ) /

∝

−

. .

PL0

280260240220200180160Temperature (K)

MAPbBr3 single crystalMAPbI3 single crystalMAPbI3 film

GaAs

Figure 3.3: Prompt photoluminescence intensity as a function of temperature. Theintensity of the prompt photoluminescence was measured in various samples (singlecrystals and thin polycrystalline films) and in a GaAs epitaxial layer, as explained in thelegend. For each sample, photoluminescence was collected from the very same spot ateach temperature. The values represent a measurement of the bimolecular recombi-nation coefficient as a function of temperature. Laser fluences in − were for theMAPbI3 single crystal, . for the MAPbBr3 single crystal, for the MAPbI3 thin film and. for the GaAs sample. The dashed lines are a guide for the eye representing power

laws with exponents− for theMAPbBr3 andMAPbI3 single crystals,− . for theMAPbI3polycrystalline film, and− for GaAs.

−− . −

43210Time (ns)

150100500Time (ns)

MAPbI3Film

150100500Time (ns)

PL in

t. (a

.u.)

MAPbBr3Single crystal

43210Time (ns)

PL in

t. (a

.u.)

43210Time (ns)

150100500Time (ns)

MAPbI3Single crystal

Figure 3.4: Photoluminescence decays as a function of temperature. The time-resolvedphotoluminescence decays in MAPbI3 and MAPbBr3 samples demonstrate that the decaydynamics is not directly correlated to the intensity of the emission, as it is due to complexnon-radiative recombination channels. Laser fluences in − were for the MAPbI3single crystal, . for the MAPbBr3 single crystal, for the MAPbI3 thin film and . forthe GaAs sample.

2

4

68

1

2

4

68

10

2

4

6

Tim

e (n

s)

280240200160Temperature (K)

CH3NH3PbI3Half-life1st momentd(PL)/dt @t=0

CH3NH3PbBr3Half-life1st momentd(PL)/dt @t=0

Figure 3.5: Decay times as a function of temperature. The decay time for photolumines-cence was extracted from themeasurements according to several definitions, as indicatedin the legend and detailed in themain text; in all instances, such times did not show a cleardependence on temperature.

=

∫∞ ( )∫∞ ( )

=

| =± %

= +

≫

= . × − −

≈

=√

( )/

≈= / ( )

( )

× −

=

= − − ≈%

= = −

− ≈

=

∝− /

=

−

−

−

450

500

550

600

650

00.2

0.4

0.6

0.81

1.2

1.4

1.6

hν

E

k Ground state

resonant X

Abs

450

500

550

600

650

00.2

0.4

0.6

0.81

1.2

1.4

1.6

hν

E

k Ground state

n on - r e s o nan t e - h X

Abs

450

500

550

600

650

00.2

0.4

0.6

0.81

1.2

1.4

1.6

hν

E

k Ground state

n on - r e s o nan t e - h

Abs

phonon

Figure4.1: Sketch of the outcomes of optical absorption in a semiconductor. Theblack lineon each panel depicts the perovskite absorption spectrum, while the blue line representthe exciton dispersion. Free electron-hole pair states are represented by the blue shad-ing. A two-particle representation is employed to show together in the same dispersioncurve excitons and free carriers; for unbound pairs, the wavevector is therefore the sumof electron and hole wavevectors. Resonant excitation just below the bandgap createsexcitons (left panel), while non-resonant excitation in the continuum as a direct processcan only lead to free electron hole couples, because of momentum conservation (centrepanel). Geminate excitons may be created with an indirect process assisted by a phonon(right panel).

Figure 4.2: Time-resolved differential photoluminescence decays from MAPbBr3. Top leftpanel: sketch of the experimental set-up, with the photoluminescence signal consistingof two contributions, the transient emission triggered by the arrival of each laser pulse,and the quasi-stationary signal due to the long-lived population of photoexcitations. Thelatter contribution becomes more and more intense for increasing repetition rate. Laserpulses, 100 fs in duration and 400 nm inwavelength, are provided by a frequency-doubledTi:sapphire oscillator with 80 MHz repetition rate. Main panel: normalized photolumi-nescence decays are shown in different colours corresponding to different excitation flu-ence; red and orange decays are produced with excitation by the MHz source, cyan andblue decays instead with the larger fluence form the kHz excitation source; at low excita-tion, the characteristic decay time is shorter than the temporal resolution of the apparatus,but it becomes longer for higher excitation fluence, as emission from geminate excitons isovercome by emission due to bimolecular recombination of free carriers; when the fluenceexceeds . − , the decay time again decreases due to faster higher-order recombi-nation. Top right inset: absorption (black) and photoluminescence spectra taken at = ,

( ), (red at low excitation, blue at high excitation).

∼

×× −

=!√ ≈ . .

! ≈

( , ) = ( , )− < ( ) ( , )

< ( )

=

=

( )

<

=

10-4 10-3 10-2 10-1 100 101

Pulse Fluence ( J/cm 2)

PL0

PL0 (kHz exc)PL0 (MHz exc)PLt=1ns (MHz exc)PLt=1ns (KHz exc)

1013 1014 1015 1016 1017 1018Injected carrier density (cm-3)

0.1

1

Dec

ay ti

me

(ns)

T=300K

m=1

m=2

m=2

Figure 4.3: Instantaneous photoluminescence as a monitor for geminate exciton creation.Main panel: Log-log plot of the instantaneous intensity of photoluminescence (filleddots, at time = , empty circles at = ns) as a function of the pulse fluence; thered markers refer to excitation with a 80 MHz Ti:sapphire oscillator (100 fs, 400 nm), theblue dots to the excitation with a larger fluence, 1 kHz regenerative amplifier (130 fs, 392nm); the dashed lines represent power dependences with exponent 1 (red) and 2 (blue andblack). Top panel: photoluminescence decay time as a function of pulse fluence (bottomaxis) or corresponding injected photoexcitation density (top axis).

=

< ¯ = − − < ¯ = × −

> ¯( > ¯ )

∝ < ¯

∝ > ¯ > ≷ ¯

=

=

=

(

!

) / (−

).

( = ) ≈× −

1011 1012 1013 1014 1015 1016 1017 1018

Injected carrier density (cm-3)

Log

PL0

0 500 1000 Time (ps)

Nor

m. P

L

530 550 570Wavelength (nm)

Nor

m. P

L10-6 10-4 10-2 100

Pulse Fluence ( Jcm2)

n>1017cm-3

n=1016cm-3

n<1015cm-3

n=1.1 1018cm-3

n=5 1016cm-3

PL0 15KPL0 80K

m=1

m=1

m=2

n=1013cm-3

Figure 4.4: Exciton crossover at . Main panel: log-log plot of the instantaneous pho-toluminescence intensity as a function of the injected carrier density (proportionalto laser fluence); solid red dots represent measurements at temperature; data col-lected at and at the highest fluences have been added for comparison. Upperleft inset. Emission spectra in three excitation regimes are shown for comparison: freecarrier regime, where geminate excitons are detected ( = , < − ); free car-rier regime where only free carriers are detected ( = , ∼ − ); exciton regime( > − ). Lower-right inset: time decay of photoluminescence in three excitationregimes, where geminate excitons are detected (blue trace, = , = − ), wherefree carriers prevail (black trace, = , = × − ) and finally when secondaryexcitons take over free carriers (red trace, = , = . × − ).

( = ) = × − −

10-5 10-4 10-3 10-2 10-1

Pulse Fluence ( J/cm 2)

cw P

L In

tens

ity

550 600 650Wavelength (nm)

Nor

m. P

L

15K

300K

m=3/2

1.4 10-6 J/cm 2

1.4 10-4 J/cm 2

10-2 J/cm 2

1.25 10-1 J/cm 2

Figure 4.5: Quasi-CWphotoluminescence. Main panel: log-log plot of the time-integratedphotoluminescence intensity as a function of the proportional to laser fluence; the powerlaw dependence as an exponent close to . , meaning that free electron-hole pairs are themajority photoexcitations in steady state. Inset: PL spectra recorded by a CCDwith a grat-ing spectrometer, at various excitation fluences, as marked.

=

−

−

= /

=

≈ ( )

≈ × −

=

√≈

∼ −

−

= ( )/ ,

=

√)

√ ,

=

≈ . − / /

∼ − /

∼ −

/

/

∝ ∝

∝ / ∝ /

∝ ∝

× −

∼ − ) /

\