Hot Carrier Cooling and Photoinduced Refractive Index
Changes in Organic-Inorganic
Lead Halide Perovskites.
Fig 1 – Carrier temperatures and cooling rates
Time Delay
(ps)
0.30
Normalized 
0.8
0.15
0.6
0.00
0.4
0.2
1.70
1.75
1.80
0.0
-0.2
𝐸f −𝐸
ΔT
[𝐸 > 1.7𝑒𝑉] ∝ 𝑒 𝑘B 𝑇c
T
-0.4
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.1
1.3
1.5
1600
Carrier Temperature (K)
1.0
N (1017cm-3)
1400
1.6
6.4
32
64
1200
1000
800
600
400
-0.6
1.6
1.8
2.0
Energy (eV)
2.2
300
600
900
1200
1500
Time (fs)
Fig 1 a.
Normalized transient absorption (TA) spectra of CH3NH3PbI3, pumped above the band-edge at 2.25eV (pump resolution ~200fs) , showing higher
carrier temperatures for early times (before 2.5ps) with a carrier density of N = 6.4 x1017 cm-3. Cooling rates were obtained by performing a global fit of
exponential decays to the high energy tail of each timeslice – see inset – between E = 1.7 eV and E = 1.85 eV. b. Carrier cooling curves at stated carrier densities
obtained using the method described in Fig1 a. At densities below ~ 6 x1017 cm-3, starting from an initial temperature of ~1000 K down to near room
temperature, the carriers cool with a principal rate constant of ~ 230 fs. At higher carrier densities there is a retardation of the cooling rate consistent with a hot
phonon bottleneck effect, and at early times (~ 400fs), the fits give a much higher absolute carrier temperature.
T/T (x10-3)
6
4
2
0
1.55
1.60
1.65
1.70
1.75
Energy (eV)
Fig 2 a.
~2x1017
1.80
1.85
1
0.8
N (1017cm-3)
1.6
6.4
32
64
0.6
1
0.8
0.6
0.4
0.4
0.2
0.2
400
600
800
1000
Normalized T/T
Pump Energy (eV)
2.06
1.9
1.71
1.63
Normalized Carrier Temperature
Fig 2 – Carrier Cooling into a Re-Normalized Band-Gap
1200
Time (fs)
Transient absorption spectra of CH3NH3PbI3 for a series of different pump energies, taken at 250fs after pump excitation, with carrier densities of
As the pump energy is moved towards resonance with the band edge, the negative feature (indicated by blue arrow) at ~1.58eV diminishes. With a
cm-3.
resonant pump energy of 1.63eV (FWHM = 25meV), the negative feature completely disappears. b. Normalized kinetics of the negative TA signal at 1.58eV
(dashed lines) compared with normalized carrier cooling kinetics (solid lines) at carrier densities stated in legend. As carrier density is increased, the TA kinetics of
the 1.58eV feature show the same trend as the carrier cooling rates - higher densities give slower decays - but the normalized cooling curves also show a
consistently faster decay than the decay of the 1.58eV TA feature.
Fig 2 – Carrier Cooling into a Re-Normalized Band-Gap
f/Tc (x10-5 ) / K-1
0
-1
-2
-3
N x1017cm-3
-4
1.6
6.4
32
64
-5
1
2
3
4
5
1/Tc2 (x10-6) / K-2
Fig 2 c.
Change in the transient absorption kinetic 𝜕𝑓 (taken at 1.58eV) divided by the change in carrier temperature 𝜕Tc , plotted versus 1 𝑇c2 for stated
carrier densities. Here we present the same data as Fig2b (before normalization), but with time plotted as an intrinsic variable. 𝜕𝑓 𝜕𝑇c is obtained by taking the
derivative of f with respect to time - 𝜕𝑓 𝜕𝑡 - and dividing this by the change in Tc with respect to time - 𝜕𝑇c 𝜕𝑡. For each carrier density, at high temperatures,
𝜕𝑓 𝜕𝑇c is proportional to -1 𝑇c2 . Linear fits are shown for each data set.
d. A schematic representation showing the relationship between carrier cooling and band-gap renormalization. Carriers are excited into/out of
conduction/valence bands, photo-excited carriers interact and reduce the band gap (from Eg0 to EgBGR) due to band-gap renormalization. Carriers thermalize to
quasi-equilibrium Fermi-Dirac distributions with effective carrier temperatures. Higher carrier temperatures occupy less states at the new band edge than cooler
carrier distributions. As carrier temperature reduces, increased occupation of the lower energy states leads to a reduction of photo-induced absorption and
eventually a positive bleach signal in the TA spectra.
Fig 3 – Refractive index changes in transient absorption
0.03
0
0.15
T/(T+R)
R/(T+R)
(T+R)/(T+R)
-2x10-3
0.10

Signal
0.02
0.01
0.00
1.6
1.8
2.0
T/Texpt.
T/Tmodel

n
Free carrier
0.05
0.00
-0.01
-0.05
1.6
Fig 3 a.
-1x10-3
1.7
1.8
Energy (eV)
1.9
2.0
1.6
1.7
1.8
1.9
2.0
Energy (eV)
Transient absorption (TA), transient reflection (TR), and ‘intrinsic’ transient absorption spectra of CH3NH3PbI3-xClx on Al2O3. The data shows a simultaneously
measured transient absorption spectrum (black line) and transient reflection spectrum (red line) with a common weighting (T+R), at a carrier density of ~5x1017 cm-3, taken at 5
ps after excitation with a 2.07 eV vertically polarized pump and a p-polarized probe beam at an incident angle of θ = 45°. The transient reflection spectrum is different in
∆𝑇+∆𝑅
spectral shape to the transient absorption. The blue line is a summation of the TA and TR spectra with a common weighting (
) and shows a rough approximation of what
𝑇+𝑅
the ‘intrinsic’ transient absorption response would be, were there no changes in reflectivity. b. Photo-induced contributions to the total transient absorption of CH3NH3PbI3.
The black circles show measured ∆T⁄T (t = 4ps, N = 6.4x1018 cm-3, θ = 90 °, 54.7° probe polarization, 2.25eV pump). The black line is the fitted model spectrum excluding
coulombic effects (mr = 0.14 me, Γ = 0.013 eV, a = 3, ∆EBGR = 5 meV). The red, green and blue lines show the components of the modelled ∆T⁄T from, respectively; changes in
absorption, ∆α – blue line , changes in refractive index, ∆n – red line line, and free carrier absorption – green line (and inset). The change in refractive index is a significant part
of the TA spectrum, however the free carrier contribution (here modelled with a scattering time constant of 20 fs) shows a negligible contribution.
Normalized T/T
17
N (10 cm )
0.8
1.6
6.4
32
64
0.6
0.4
0.2
0.0
0.05
Experiment
Model
0.04
0.03
0.1
0.02
0.01
0.00
0.0
-0.2
1.6
Fig 4 a.
-3
T/T maximum value
0.2
1.0
1.7
Energy (eV)
1.8
0
2
4
6
8
N (1018 cm-3)
Global fit of transient absorption model to experiment for a range of carrier densities (stated in inset legend). Normalized experimental ∆T⁄T of CH3NH3PbI3 on
spectrosil is shown by open circles (t = 4ps, θ = 90 °, 54.7° probe polarization, 2.25eV pump). The solid lines show the model ∆T⁄T response for the corresponding carrier
densities. The model presented here is that described in Fig3b. The parameters are mr = 0.14 me, Γ = 0.013 eV, a = 3, C = 0.58 m-1s-1/2.. The model captures the Burstein-Moss
spectral broadening (controlled by the reduced effective mass) with increasing carrier density. b. Carrier density dependence of spectral maxima and minima. Experimental
maxima (red squares) and absolute value of spectral minima (bright green squares) are compared to the model of Fig 4a (thin dark red and dark green lines for simulated peak
maximum and minimum, respectively). The model captures the peak ∆T⁄T saturation behaviour. This saturation is also governed by the effective mass.
|T/T minimum value|
Fig 4 – Effective Mass calculation from reflection corrected transient absorption spectra
Supplementary
0.002
0.001
T/T
0.000
-0.001
Pump Energy (eV)
1.90
1.71
1.65
-0.002
-0.003
-0.004
-0.005
0.1
1
10
100
1000
Time (ps)
Figure S2 Cooling rate differences for different excitation energies. Kinetic traces of 1.58eV negative feature of transient absorption
spectra (as proxy for carrier cooling rate) at a range of pump energies, with fluences corresponding to a carrier density of approximately N = 2x1017 /cm3. Kinetic
traces of 1.58eV feature illustrate the biexponential nature of its rise time, and the disappearance of the fast rise component when pumped close to resonance.
Complex Refractive Index
2.4
2.2
2.0
1.8
1.6
Refractive Index
Extinction Coefficient
1.4
1.2
1.0
0.8
0.6
0.4
0.2
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
Energy (eV)
Figure S4In-plane complex refractive index of CH3NH3PbI3.
Ellipsometric measurement is described in methods section.
0.05
()
0.005
0.005
50
40
30
20
0.03

RR
0.010
()
0.04
50
40
20
0.02
0.000
RR
0.015
polarization
-0.005
0.000
0.01
-0.005
0.00
P
S
-0.010
-0.01
1.6
1.7
1.8
1.9
1.5
Energy (eV)
1.6
1.7
1.8
1.9
2.0
2.1
Energy (eV)
S 5 a. Angle dependence of ∆R/R spectra of CH3NH3PbI3-xClx on Al2O3. N = 5x1017 cm-3, taken at 5 ps after
excitation with a 2.07 eV vertically polarized pump and s-polarized probe. Incident angles are as stated in legend.
b. Angle dependence of ∆T/T spectra of CH3NH3PbI3-xClx on Al2O3 corresponding to the reflections measured in S5a.
c. Probe polarization dependence of ∆R/R spectra of CH3NH3PbI3-xClx on Al2O3. N = 5x1017 cm-3, taken at 5 ps after
excitation with a 2.07 eV vertically polarized pump.
1.6
1.7
1.8
Energy (eV)
1.9
0.03
Signal
0.02
Experiment
T/(T+R)
R/(T+R)
0.01
Model
T/(T+R)
R/(T+R)
0.00
-0.01
1.6
1.7
1.8
1.9
2.0
Energy (eV)
S 7. Transient transmission and reflection spectra of CH3NH3PbI3-xClx on Al2O3 – experiment and model.
Simultaneously measured transient transmission spectrum (black circles) and transient reflection spectrum (red
circles) of the sample at a carrier density of ~5x1017 cm-3, taken at 5 ps after excitation with a 2.07 eV vertically
polarized pump and a p-polarized probe beam at an incident angle of θ = 45° to the sample. The solid lines are a fit to
the experimental data using the simple band-filling model and reflections calculated from the angular dependent
Fresnel equations. Fit parameters: Eg = 1.64 eV, mr = 0.15 me, Γ = 0.03 eV, a = 1.5.
Tc (K)
1.0
Time delay (fs)
1000
900
800
700
600
500
400
300
0.4
0.2
250
300
350
400
500
2500
1.5
(10-3)
0.6
T/T
0.8
Normalized 
2.0
0.0
-0.2
1.0
Expt.
0.5
-0.4
0.0
-0.6
-0.8
-0.5
-1.0
1.6
1.7
1.8
1.9
Energy (eV)
2.0
2.1
1.6
1.7
1.8
Energy (eV)
S 8 a. Effect of carrier temperature on model TA spectra. Model parameters; N = 6.4x1018 cm-3 , mr = 0.14 me, Γ
= 0.013 eV, C = 0.58 m-1s-1/2 , ∆EBGR = 15 meV. Temperatures stated in legend. The figure shows how the model
recreates the sub-band-gap TA feature and spectral broadening for high carrier temperatures and a moderate
bandgap renormalization. b. Early time TA spectra of CH3NH3PbI3 with overlayed model TA spectra. Experimental
spectra obtained with 2.25eV pump, N =1.3x1017 cm-3. The model fits were obtained by applying the fit described
in Fig 4c (parameters; mr = 0.14 me, Γ = 0.013 eV, a = 3, C = 0.58 m-1s-1/2, B = 1.2x104 ) assuming Tc = 300 K to the
fully cooled spectra at 2500fs. Then for the earlier, hotter spectra, the model was fitted to experiment allowing
only Tc and C as free parameters.
250
300
350
400
500
2500
Model
1
Exponential fit for Tc
Exponential fit for Tc
Model fit for Tc
Model fit for Tc
800
Normalized c
Carrier Temperature (K)
900
700
600
500
400
300
0.1
500
1000
1500
Time Delay (fs)
2000
2500
400
600
Time (fs)
S 8 c . Carrier cooling curves of CH3NH3PbI3, for the data described in S()b. The blue circles show the carrier
temperatures obtained by the method and fits shown in S()b. The black squares show carrier temperatures
obtained by the method described in Fig 1. The cooling curves show a very similar trend, differing by a constant
temperature offset. d. The data of S()c normalized and presented on a vertical log scale plot. The overlapping
temperature decays show that the two methods for obtaining carrier cooling give very similar cooling rates.
800
0.6
mr (me)
T/T
0.08
0.1
0.12
0.16
0.2
0.4
T/T
0.2
mr (me)
0.2
0.08
0.1
0.14
2
0.0
0.0
1.7
1.8
Energy (eV)
1.9
0
5
10
15
20
18
25
30
-3
N (10 cm )
S 9a. Effect of reduced effective mass on modelled transient absorption spectrum assuming no coulomb
interaction. Each simulation has N = 6.4x1018 cm-3 , Γ = 0.013 eV, a = 3, C = 0.58 m-1s-1/2 , ∆EBGR = 5 meV. Effective
masses are those stated in legend. The figure highlights how sensitive the final spectral shape (peak position and
FWHM) is on mr. b. Effect of effective mass on ∆T/T kinetics. Parameters are held constant – those stated in S()a.
– and effective masses are stated in legend.
35
EBGR(meV)
0
5
25
1
2
4
6
0.1
0.0
0.1
T/T
T/T
Asymmetry, a
0.0
1.6
1.7
1.8
1.9
Energy (eV)
2.0
2.1
1.6
1.7
1.8
1.9
Energy (eV)
S 9c. Effect of effective mass asymmetry parameter, a, on modelled transient absorption spectrum. Each
simulation has N = 6.4x1018 cm-3 , mr = 0.14 me, Γ = 0.013 eV, C = 0.58 m-1s-1/2, ∆EBGR = 5 meV. Asymmetry
parameters are those stated in legend. The primary effect of a is to modify the curvature of the reflection induced
high energy negative feature. d. Effect of bandgap renormalization on modelled spectral shape of transient
absorption response.( N = 6.4x1018 cm-3 , mr = 0.14 me, Γ = 0.013 eV, C = 0.58 m-1s-1/2). Band gap shift stated in
legend. At low carrier temperatures the band gap shift mostly lowers and slightly sharpens the spectral peak. The
band gap shift is more apparent in the spectra at higher carrier temperatures – see S().
2.0
2.1
68.75
Experiment
Linear Fit
62.50
(meV)
56.25
50.00
43.75
37.50
0.0
0.5
1.0
1.5
N
2/3
2.0
-12
2.5
3.0
3.5
-2
(10 cm )
S 11. Burstein-Moss shift for reduced effective mass calculation, without refractive index changes. The half-width at
full maximum of the spectra is plotted against N2/3. A linear fit gives a gradient of
(Efc-Efv)-Eg (eV)
0.1
0.0
-0.1
0.1
1
18
10
-3
N (10 cm )
S 12. Difference in quasi-Fermi energies of conduction and valence bands minus the bandgap, versus carrier
density, according to simple band-filling model with parameters described in Fig4c. The threshold for optical gain
is the point at which (Efc-Efv)-Eg becomes greater than zero. The dashed red line indicates that this transition is
predicted to occur at N ~ 2.5x1018 cm-3.
Fig S4 – Models of absorption and change in absorption
1.0x105
 (cm-1)
8.0x104
6.0x104
Discrete states
Continuum states
Disc. + Cont.
Experiment
4.0x104
2.0x104
0.0
1.50
1.75
2.00
Energy (eV)
Fig S13a.
Steady state absorption of a (65nm) film of CH3NH3PbI3 on spectrosil – thin black line – taken by reflection-corrected UV-vis and photothermal deflection
spectroscopy. The bright green line is a fit to the experimental data according to Elliot’s Wannier exciton theory, broadened with a pseudo-Voigt profile. Exciton binding energy
εB= 17 meV, Eg = 1.645 eV, Γ = 0.045 eV. The lorentzian portion of the broadening profile was given a weighting of 0.001. The broadening parameters were obtained by a
comparison of the band tail of the model to the experimental data over three orders of magnitude (see Fig S 4b). The blue line shows the contribution to the model spectra
from continuum states, and the red line shows the contribution from discrete states (Wannier excitons).
 (cm-1)
105
Discrete states
Continuum states
Disc +Cont
Experiment
104
103
1.6
1.8
2.0
Energy (eV)
S 13b
Figure 13a with a log scale y-axis. For nearly 3 orders of
magnitude in α the band tail shows purely gaussian (inhomogeneous)
broadening. The model shows good agreement with the experimental
data over this range. The PDS absorption data was spliced together
with the scatter-free UV-Vis at α = 5x104 cm-1.
(cm-1)
2.0x104
Model used
Discrete
Continuous
Disc + Cont
No Enhancement
1.5x104
1.0x104
5.0x103
0.0
1.6
1.7
1.8
1.9
2.0
Energy (eV)
S14 Models of change in absorption coefficient due to the presence of photo-excited states. The solid red line is a
model for the change in absorption coefficient that includes band-filling effects, coulomb enhancement and
reduction in oscillator strength due to screening and/or occupation of excitonic transitions. Carrier density N = 6x1018
cm-3, reduced mass, mr = 0.14 me, εB= 17 meV, Γ = 0.04 eV, effective mass asymmetry a = 1.9. The red dashed line
shows the contribution to this model due to change in the continuum states, and the dotted line shows the modelled
contribution due to reduction of the excitonic transition. The black curve is a change in absorption coefficient with a
simpler model that neglects coulombic effects. mr = 0.14 me, Γ = 0.03 eV, a = 3. We see that for higher energies,
above ~1.75 eV, where the signal is dominated by band-filling and Elliot’s absorption theory converges towards a
simple 𝐸1 𝐸 − 𝐸𝑔 dependence, the non-excitonic model broadly agrees with the model that includes a coulomb
enhancement.
T/T
0.15
0.10
Experiment
Model
0.05
0.00
1.6
1.7
1.8
1.9
Energy (eV)
S 15. Un-normalized fits and experiment
2.0
2.1
Normalized T/T
1.0
N (1017cm-3)
0.8
1.6
6.4
32
64
0.6
0.4
0.2
0.0
-0.2
1.6
1.7
1.8
1.9
2.0
Energy (eV)
S16. Fitted spectra including coulomb enhancement and interference