Arman Mahboubi Soufiani 1 , Fuzhi Huang 2 , Peter Reece 3 ,
Rui Sheng 1 , Anita Ho-Baillie 1 and Martin A. Green 1*
1
Australian Centre for Advanced Photovoltaics,
School of Photovoltaic and Renewable Energy Engineering,
University of New South Wales, Sydney, NSW 2052, Australia
2
Department of Materials Engineering, Monash University, Clayton, Vic 3800, Australia
3 School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
Matlab R2012b software was used for all the fitting purposes. In order to minimize the chance of trapping in any local minima we primarily covered the full parameter space by using the multistart solver – as the global search function – with lsqcurvefit as the local solver. The optimum parameters obtained in this way were used as input to the nlinfit function, using Levenberg-Marquardt nonlinear least squares algorithm, to extract the final optimum fitting parameters and the corresponding 95% confidence levels.
In order to apply a consistent approach to all the fittings to the experimental optical density/absorption coefficient spectra and to lessen the effect of valence- and conduction-band non-parabolicity expected in organic-inorganic perovskites 1 , we have fitted the data within one binding energy plus a multiple of Γ (i.e. HWHM of the
Lorentzian broadening function) above the band gap or parametrically,
E g

[ Ry
0
 a
 
] . However, assigning 1.0, 1.25 and 1.5 to a did not affect the fitting parameters noticeably, in particular, the exciton binding energy. As such, all the results presented in this work is with the multiplier, a , equal to 1.0.
- 1 -

Pseudo-Voigt profile composed of weighted sum of Lorentzian (L) and Gaussian (G) profiles with a common full width at half maximum (FWHM) is used to convolve with the theoretical spectrum (Eq. 1 of main text) for fitting purposes
2
.Analytical approximations to the convolutions used in these fitting are:
 
G

AR
0
2
1
E 
2 R
0 n
N 

1 e

( E

E g

R
0
/ n
2
)
2 n
3
2
 
2
 2
 



1

Erf 
E
2
E g




2

 e

( E

E g

R
0
)
2
58 2

2
 2

( E

E g

116 R
0
R
0
)




1

Erf
E

E g

R
0
2






S1 for the Gaussian and,
 
L


116 R
0

AR
0
2
1
E ln
2 R
0

ArcTan


E

E g


R
0 n
N 

1


( E
( E

E g


/
R
0 n
3
/ n
2
)
2

E g
58


R
0
R
0
)

  2
ArcTan




E



0 .
5



E m


ArcTan




( E
58


E
E m
R
0
)




E g



( E m





E g
116 R
0

R
0
)




( E

( E
E g


R
E m
) 2
0
) 2



2

2



S2 for the Lorentzian profile. We confirmed that a maximum of 100 discrete excitonic states
(i.e. n =100) suffice, above which changes in the theoretical spectrum are not apparent.
We also investigated improving the fits to the experimental spectra by following
Toyozawa’s derivation of exciton absorption line-shape 3
in which the broadening associated with each excitonic state in the hydrogen-like line series and that of the continuum absorption portion differ. Commonly, an asymmetry parameter (

1 ) accounting for exciton discrete states interaction with the continuum of states is also introduced, which we have not included in the line-shape modelling. We also investigated scaling the HWHM of the excited excitonic states
4
:
- 2 -


X
( n )  
C


C
 
X
( 1 ) n
2
      S3 where
 ( 1 )
X
,

C
,
 ( n
X
) and n are the HWHM of the exciton ground state, continuum of states and the excited excitonic states and the principle quantum number, respectively.
The first two are independent free fitting parameters. We note that this relation between the damping parameters of the states is an empirical one giving reasonable
fits with acceptable physical parameters for III-V and II-VI semiconductors
4, 5
. This
relation indicates that the damping parameter increases with energy (i.e. quantum number) of the states.
To calculate the binding energy of the 1S excitons created in a polar semiconductor in interaction with a cloud of virtual phonons (i.e. polaronic exciton problem), we adapted the variational functional approach of Kane
6
applicable to weak- and intermediate-coupling modes. Kane’s formalism is a modified version of the Pollman-
Büttner (PB) 7 perturbation theory method in which renormalization of the optical frequency dielectric constant and bare effective masses for static dielectric constant and polaronic masses is accounted for, respectively, in the weak-coupling regime, whereas, PB only considered the former renormalization.
Critical basic physical parameters of the material influencing the generated excitonic binding energy in such medium are the effective mass ratio of the carriers, permittivity with and without ionic contributions (
 ion and


respectively) and relevant longitudinal optical phonon energy. Due to the uncertainty associated with the calculated/measured above-mentioned parametersin the literature (provided in
Table S1), regions of plausible exciton binding energy values are calculated. To calculate the exciton binding energy and its corresponding single effective LO phonon energy, subsequent to finding the appropriate electron to hole bare effective mass ratio in Table I of Kane, we calculated the unknown parameter in Eq. 47
6
through which the binding energies can be directly calculated using Eq. 46. Thereafter, we have calculated the corresponding LO phonon energies from Eq. 51
6
.
- 3 -

Table S1 References to the physical parameters used in Figure 2a of main text.


 ion 𝒎 ∗ 𝒆 and 𝒎 ∗ 𝒉
Upper Bound Theory 8 Experiment 9 Theory 10
CH
3
NH
3
PbI
3
Lower Bound
Experiment
11, 12
Experiment
13
Theory
1
Upper Bound Theory
14
Experiment
13
Theory
14
CH
3
NH
3
PbBr
3
Lower Bound Experiment 11 Experiment 13
Scaled according to lower bound of carrier mass ratio of triiodide
A four phonon fit to Eq. (3) of the text gave a similarly good fit to the best 5-phonon f it found. From these fitting parameters, the effective LO energy can be calculated.
Table S2 Optimum fitting parameters (THz) in Eq. (3) of the main text used to fit the experimental permittivity data presented in Figure 2b.
Modes

LOi

TOi

Li

Ti

Ld

Td
1 0.9054 0.0035 2.5645 2.4438
2
3
4
1.0476 0.9969 0.2412 0.2768
1.2066 1.0476 2.0538 0.8349
0.0722 0.0275
1.2950 1.2066 1.1620 0.7028
- 4 -

Figure S1 demonstrates the quality of fit to the measured absorption coefficient compared to values half and twice the optimal fitting value.
Fig. S1 Absorption coefficient spectrum band-edge decomposition and fit assessment at room temperature.
(a) Absorption coefficient of CH
3
NH
3
PbI
3
(grey open square) was measured using spectroscopy ellipsometry at room temperature. Black solid line shows the actual fit to the band-edge with effective Rydberg of 11.1±0.1 meV. Blue short dash and the orange dash-dot dot lines illustrate force-fitted theoretical spectrum by Ry
0
of 5.0 meV and
24 meV, respectively. These values are half and double that of the optimum effective exciton binding energy. The FWHM and band gap are set as free parameters in the force-fits.
Figure S2 shows the effect of sample storage in a nitrogen purged glovebox upon the extracted exciton binding energy values. Also demonstrated is the insensitivity to use of absolute absorption coefficient values or optical density data.
- 5 -

Fig. S2 Temperature-dependent exciton binding energy of CH
3
NH
3
PbI
3
measured on different substrates and at various times after fabrication. The exciton binding energies are obtained from optical density at various temperatures and from absorption coefficient only at room temperature. The perovskite film was deposited on two different types of glasses namely, soda-lime and normal microscope glass. In addition, exciton binding energies were collected at various times after sample deposition. The RT mean value is presented by red star symbol and the 95% confidence interval is calculated via student’s t-test.
Figure S3 shows the deconvolution of the fitted absorption coefficient into ground state, excited state and continuous contributions. A systematic deviation was observed at energies where excited states contributed, with the fit able to be improved using more complex models. However, these were not implemented in the main text due to the lack of a sound physical basis and the relatively small impact upon the extracted binding energy.
- 6 -

Fig. S3 CH
3
NH
3
PbBr
3 room temperature absorption coefficient spectrum band-edge decomposition and fit assessment.
The actual theory is homogeneously broadened with p-
Voigt profile (black solid line) with its decomposed discrete states and continuum of states components illustrated. The effective exciton binding energy, FWHM and band gap are Ry
0
= 37.2 meV, 2

= 72.0 meV and E g
= 2396.2±1.4 meV. Blue solid line is the theory convolved with p-Voigt profile and further modified using scaled broadening of the states as explained in Eq. S3. The optimum FWHM values for the ground excitonic and continuum of states – as the free parameters – are 2
 ( 1 )
X
= 66.9 meV and 2

C
= 50.1 meV, respectively.
Effective exciton binding energy and electronic band gap are Ry
0
= 39.6 meV and E g
=
2397.5±1.4 meV, respectively.
The Lorentzian contribution to the pseudo-Voigt broadening for the results shown in
Fig. 3 of the main text are shown in Fig. S4 below.
- 7 -

Fig.S4 Lorentzian contribution to the absorption spectrum broadening.
The increasing contribution of the Lorentzian broadening profile – or the corresponding deceasing contribution of the Gaussian broadening – with temperature to the pseudo-Voigt profile by which the theoretical spectrum is smeared is illustrated for CH
3
NH
3
PbI
3
and CH
3
NH
3
PBr
3 perovskites. The data are presented for the room temperature tetragonal phase of
CH
3
NH
3
PbI
3 and tetragonal and cubic phases’ temperature range ( ≳ 140°C) of
CH
3
NH
3
PbBr
3
.
The broadening observed in the present work for pure iodide specimens was appreciably smaller than observed in earlier work, leading to more prominent excitonic features. The broadening was instead quite similar to that seen earlier in better quality Cl doped samples.
- 8 -

Fig.S5 Material quality. Optical density spectrum at 40K of CH
3
NH
3
PbI
3
(red) and
CH
3
NH
3
PbI
3-X
Cl
X
(orange) reported by D’Innocenzo et al.
15 and CH
3
NH
3
PbI
3
(brown) used in this study are presented. The shifted spectrum of the CH
3
NH
3
PbI
3-x
Cl x
(purple) towards lower energies is to compare its spectrum broadening with that of the CH
3
NH
3
PbI
3
. The comparison shows the high quality of the CH
3
NH
3
PbI
3 material used in this work with much smaller spectrum broadening than that of CH
3
NH
3
PbI
3 reported by D’Innocenzo et al. in [15]
(i.e. indicative of its higher crystallinity). The broadening observed for the CH
3
NH
3
PbI
3 material in this work is also comparable to that of the CH
3
NH
3
PbI
3-X
Cl
X
in ref.
15
.
- 9 -

Fig.S6 MAPbI
3
film morphology. The top-view SEM image of MAPbI
3
film illustrating grains of 300-500 nm in size.
1. Brivio F, Butler KT, Walsh A, van Schilfgaarde M. Relativistic quasiparticle self-consistent electronic structure of hybrid halide perovskite photovoltaic absorbers. Physical Review B
89, (2014).
2. Di Rocco H, Cruzado A. The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio. Acta Physica Polonica A
122, 666-669 (2012).
3. Toyozawa Y. Theory of line-shapes of the exciton absorption bands. Progress of Theoretical
Physics 20, 53-81 (1958).
4. Le Toullec R, Piccioli N, Chervin J. Optical properties of the band-edge exciton in GaSe crystals at
10 K. Physical Review B 22, 6162 (1980).
5. Passler R, Griebl E, Riepl H, Lautner G, Bauer S, Preis H, Gebhardt W, Buda B, As D, Schikora D.
Temperature dependence of exciton peak energies in ZnS, ZnSe, and ZnTe epitaxial films.
Journal of applied physics 86, 4403-4411 (1999).
6. Kane EO. Pollmann-Büttner variational method for excitonic polarons. Physical Review B 18,
6849 (1978).
7. Pollmann J, Büttner H. Effective Hamiltonians and bindings energies of Wannier excitons in polar semiconductors. Physical Review B 16, 4480 (1977).
8. Filippetti A, Delugas P, Mattoni A. Methylammonium Lead-Iodide Perovskite: Recombination and
Photoconversion of an Inorganic Semiconductor Within a Hybrid Body. The Journal of
Physical Chemistry C (2014).
9. Wehrenfennig C, Liu M, Snaith HJ, Johnston MB, Herz I. Charge-Carrier Dynamics in Vapour-
Deposited Films of the Organolead Halide Perovskite CH3NH3PbI3− xClx. Energy &
Environmental Science (2014).
10. Giorgi G, Fujisawa J-I, Segawa H, Yamashita K. Small photocarrier effective masses featuring ambipolar transport in methylammonium lead iodide perovskite: A density functional analysis. The Journal of Physical Chemistry Letters 4, 4213-4216 (2013).
- 10 -

11. Hirasawa M, Ishihara T, Goto T, Uchida K, Miura N. Magnetoabsorption of the lowest exciton in perovskite-type compound (CH3NH3)PbI3. Physica B: Condensed Matter 201, 427-430
(1994).
12. Lin Q, Armin A, Nagiri RCR, Burn PL, Meredith P. Electro-optics of perovskite solar cells.
Nature Photonics (2014).
13. Poglitsch A, Weber D. Dynamic disorder in methylammoniumtrihalogenoplumbates (II) observed by millimeter ‐ wave spectroscopy. The Journal of Chemical Physics 87, 6373-6378 (1987).
14. Le Bahers T, Labat F, Sautet P, Melissen S. Electronic properties of PbX3CH3NH3 (X= Cl, Br, I) compounds for photovoltaic and photocatalytic applications. Physical Chemistry Chemical
Physics (2014).
15. D’Innocenzo V, Grancini G, Alcocer MJP, Kandada ARS, Stranks SD, Lee MM, Lanzani G,
Snaith HJ, Petrozza A. Excitons versus free charges in organo-lead tri-halide perovskites. Nat
Commun 5, (2014).
- 11 -