Supplementary Information
Polaronic Exciton Binding Energy in Iodide and
Bromide Organic-Inorganic Lead Halide Perovskites
Arman Mahboubi Soufiani1, Fuzhi Huang2, Peter Reece3,
Rui Sheng1, Anita Ho-Baillie1 and Martin A. Green1*
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
S1
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
Fitting Procedure
S1.1 Software
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
lsqcurvefitas 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.
S1.2 Consistent fitting approach
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,
Eg  [ 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-
S2 Elliot’s Theory for Absorption Spectrum Band-edge
Modelling
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:
1
2
AR0
 G 
E

e
 ( E  E g  R0 / n 2 )

N
e

2 R0 
n3 2 
n 1


 ( EE g R 0 ) 2
2 2
58 2 
2
2 2

 1  Erf

( E  E g  R0 ) 

 1 Erf
116 R 0

 E  Eg

 2 


 2


 E  E g  R0


2



 
  

S1
for the Gaussian and,
1
2
AR  2R
 L  0  0
E  
N

n 1
 E  E g  R0
ArcTan



 E  Eg
 / n3
 0.5  ArcTan
2 2
2
( E  Eg  R0 / n )   
 
 
  
 
 ( E  E g  R0 )
 E  E m   ( E  E m )  ( E m  E g  R0 )

 
 ArcTan

 
58 R0

116 R0

  58 R0 

 ( E  E g  R0 ) 2   2  


ln 

2
2
116 R0
 ( E  E m )  
 

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)

n2





S3
where X(1) , C , X(n ) 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.
S3
Exciton Binding Energy Calculation in a Polar Medium
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 PollmanBüttner (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.
Upper Bound

 ion
𝒎∗𝒆 and 𝒎∗𝒉
Theory 8
Experiment 9
Theory 10
Experiment
11, 12
Experiment 13
Theory 1
Theory 14
Experiment 13
Theory 14
Experiment 13
Scaled according
to lower bound
of carrier mass
ratio of triiodide
CH3NH3PbI3
Lower Bound
Upper Bound
CH3NH3PbBr3
Lower Bound
S4
Experiment 11
Permittivity Fitting Parameters
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
1
0.9054
0.0035
2.5645
2.4438
2
1.0476
0.9969
0.2412
0.2768
3
1.2066
1.0476
2.0538
0.8349
4
1.2950
1.2066
1.1620
0.7028
-4-
Ld
Td
0.0722
0.0275
S5
Quality of Fit to Absorption Coefficient
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 CH3NH3PbI3 (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.
S6
Effect of Sample/Perovskite Film Storage
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 CH3NH3PbI3 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.
S7
Excited Excitonic State Contributions
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 CH3NH3PbBr3 room temperature absorption coefficient spectrum band-edge
decomposition and fit assessment.The actual theory is homogeneously broadened with pVoigt 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 Eg = 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X(1) = 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.
S8
Lorentzian Broadening Fraction
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 CH3NH3PbI3 and CH3NH3PBr3
perovskites. The data are presented for the room temperature tetragonal phase of
CH3NH3PbI3 and tetragonal and cubic phases’ temperature range (≳140°C) of
CH3NH3PbBr3.
S9
Effect of Material Quality
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 CH3NH3PbI3 (red) and
CH3NH3PbI3-XClX (orange) reported by D’Innocenzo et al.15 and CH3NH3PbI3 (brown) used
in this study are presented. The shifted spectrum of the CH3NH3PbI3-xClx (purple) towards
lower energies is to compare its spectrum broadening with that of the CH3NH3PbI3. The
comparison shows the high quality of the CH3NH3PbI3 material used in this work with much
smaller spectrum broadening than that of CH3NH3PbI3 reported by D’Innocenzo et al. in [15]
(i.e. indicative of its higher crystallinity). The broadening observed for the CH3NH3PbI3
material in this work is also comparable to that of the CH3NH3PbI3-XClX in ref. 15.
-9-
Fig.S6 MAPbI3 film morphology. The top-view SEM image of MAPbI3 film illustrating
grains of 300-500 nm in size.
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