Hot Carrier Cooling and Photo-induced Refractive Index

Changes in Organic-Inorganic Lead Halide Perovskites.

Fig 1 – Carrier temperatures and cooling rates

300 600 900 1200 1500

400

600

800

1000

1200

1400

1600N (1017cm-3)

Car

rier

Tem

per

atu

re (

K)

Time (fs)

1.6 6.4 32 64

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.

1.6 1.8 2.0 2.2

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-0.4

-0.2

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1.0

No

rma

lize

d

Energy (eV)

0.1 0.2 0.3 0.4 0.5 0.7 0.9 1.1 1.3 1.5

Time Delay (ps)

1.70 1.75 1.80

0.00

0.15

0.30

ΔTT

[𝐸>1.7 𝑒𝑉 ]∝𝑒𝐸 f −𝐸𝑘B𝑇c

Fig 2 a. Transient absorption spectra of CH3NH3PbI3 for a series of different pump energies, taken at 250fs after pump excitation, with carrier densities of ~2x1017 cm-3. 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

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.

1.55 1.60 1.65 1.70 1.75 1.80 1.85

0

2

4

6

T/T

(x1

0-3)

Energy (eV)

2.06 1.9 1.71 1.63

Pump Energy (eV)

400 600 800 1000 1200

0.2

0.4

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1

0.2

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1N (1017cm-3)

1.6 6.4 32 64

No

rmal

ized

T

/T

No

rmal

ized

Car

rier

Tem

per

atu

re

Time (fs)

Fig 2 – Carrier Cooling into a Re-Normalized Band-Gap

Fig 2 c. Change in the transient absorption kinetic (taken at 1.58eV) divided by the change in carrier temperature , plotted versus for stated carrier densities. Here we present the same data as Fig2b (before normalization), but with time plotted as an intrinsic variable. is obtained by taking the derivative of f with respect to time - - and dividing this by the change in Tc with respect to time -. For each carrier density, at high temperatures, is proportional to -. 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 Eg

0 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.

1 2 3 4 5

-5

-4

-3

-2

-1

0

1.6 6.4 32 64

f/Tc

(x10

-5 )

/ K-1

1/Tc2 (x10-6) / K-2

N x1017cm-3

Fig 2 – Carrier Cooling into a Re-Normalized Band-Gap

Fig 3 – Refractive index changes in transient absorption

Fig 3 a. 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 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 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.

1.6 1.7 1.8 1.9 2.0-0.05

0.00

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0.15

Energy (eV)

T/T expt. T/T model n Free carrier

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-1x10-3

0

1.6 1.7 1.8 1.9 2.0

-0.01

0.00

0.01

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0.03S

ign

al

Energy (eV)

T/(T+R) R/(T+R) (T+R)/(T+R)

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0.4

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0.8

1.0

Energy (eV)

1.6 6.4 32 64

No

rmal

ized

T

/T N (1017cm-3)

Fig 4 a. 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.

0 2 4 6 8

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T/T

ma

xim

um

va

lue

N (1018 cm-3)

|T

/T m

inim

um

va

lue

|

Experiment Model

Fig 4 – Effective Mass calculation from reflection corrected transient absorption spectra

Supplementary

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.

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-0.005

-0.004

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T/T

Time (ps)

1.90 1.71 1.65

Pump Energy (eV)

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Com

plex

Ref

ract

ive

Inde

xEnergy (eV)

Refractive Index Extinction Coefficient

Figure S4In-plane complex refractive index of CH3NH3PbI3.Ellipsometric measurement is described in methods section.

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.

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RR

Energy (eV)

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( )

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( )

Energy (eV)

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Energy (eV)

P S

polarization

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ModelExperiment

T/(T+R) R/(T+R)

T/(T+R) R/(T+R)

Sig

nal

Energy (eV)

S 7. Transient transmission and reflection spectra of CH3NH3PbI3 x‑ Clx 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.

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2.0

Model

250 300 350 400 500 2500

T/T

(10

-3)

Energy (eV)

250 300 350 400 500 2500

Time delay (fs)

Expt.

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.

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Time (fs)

Exponential fit for Tc

Model fit for Tc

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Model fit for Tc

Car

rier

Tem

pera

ture

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)

Time Delay (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.

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Energy (eV)

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mr (m

e)

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mr (m

e)

T/T

N (1018cm-3)

0.08 0.1 0.14 2

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.

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T

/T

Energy (eV)

1 2 4 6

Asymmetry, a

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T/T

Energy (eV)

0 5 25

EBGR

(meV)

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().

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37.50

43.75

50.00

56.25

62.50

68.75

Experiment Linear Fit

(m

eV)

N2/3 (10-12cm-2)

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

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0.1

N (1018cm-3)

(Efc-E

fv)-

Eg

(eV

)

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.

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4.0x104

6.0x104

8.0x104

1.0x105

(

cm-1)

Energy (eV)

Discrete states Continuum states Disc. + Cont. Experiment

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).

Fig S4 – Models of absorption and change in absorption

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.

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103

104

105

(

cm-1)

Energy (eV)

Discrete states Continuum states Disc +Cont Experiment

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1.5x104

2.0x104

(c

m-1)

Energy (eV)

Discrete Continuous Disc + Cont No Enhancement

Model used

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 dependence, the non-excitonic model broadly agrees with the model that includes a coulomb enhancement.

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Experiment Model

T/T

Energy (eV)

S 15. Un-normalized fits and experiment

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N (1017cm-3)

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/T

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S16. Fitted spectra including coulomb enhancement and interference