Table 1 Three nominal contributions to the size-dependent experimental linewidth of MAPbBr3 CNCs, expressed in \({{{{{\rm{nm}}}}}}\) value.

From: Semi-empirical approach to assess externally-induced photoluminescence linewidth broadening of halide perovskite nanocrystals with particle-size distribution

\({D}_{{{{{{\rm{p}}}}}}}\) \(({{{{{\rm{nm}}}}}})\)

\({R}_{{{{{{\rm{p}}}}}}}\) \(({{{{{\rm{nm}}}}}})\)

\({{{{{\boldsymbol{\lambda}}}}}}_{{{{{{\rm{p}}}}}}}\) \(({{{{{\rm{nm}}}}}})\)

\({{{{{\boldsymbol{\Gamma}}}}}}_{E{{{{{\rm{XP}}}}}}}\left({{{{{\boldsymbol{\lambda}}}}}} \right)\) \(\left({{{{{\rm{nm}}}}}}\right)\)

\({{{{{\boldsymbol{\Gamma}}}}}}_{{{{{{\rm{SD}}}}}}}^{{{{{{\rm{ext}}}}}}}\left({{{{{\boldsymbol{\lambda}}}}}} \right)\) \(\left({{{{{\rm{nm}}}}}}\right)\)

\({{{{{\boldsymbol{\lambda}}}}}}_{{{{{{\rm{QC}}}}}}}\left({{{{{\boldsymbol{\lambda}}}}}} \right)\) \(\left({{{{{\rm{nm}}}}}}\right)\)

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left({{{{{\boldsymbol{\lambda}}}}}} \right)\) \(\left({{{{{\rm{nm}}}}}}\right)\)

\({{{{{\boldsymbol{\omega}}}}}}_{{{{{{\rm{p}}}}}}}\) \(\left(\times {10}^{15}{s}^{-1}\right)\)

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{EXP}}}}}}}\left({{{{{\boldsymbol{\omega}}}}}} \right)\) \(\left(\times {10}^{15}{s}^{-1}\right)\)

2.7

1.35

475

35.9

35

1.9

-1.0

3.966

0.293

3.2

1.6

498

33.7

32

1.7

0

3.783

0.256

5.0

2.5

514

28.9

18.5

1.1

9.3

3.665

0.206

6.0

3.0

518

26.4

14.5

0.9

11.0

3.637

0.185

7.5

3.75

522

22.2

6.0

0.8

15.4

3.609

0.154

8.7

4.35

523

22.3

5.4

0.7

16.2

3.602

0.154

9.9

4.95

524

23.4

4.5

0.6

18.3

3.595

0.161

19.5

9.75

525.1

23.5

2.2

0.3

21.0

3.587

0.161

22.5

11.25

526.0

24.7

2.2

0.3

22.2

3.582

0.168

23.7

11.85

526.2

25.5

2.2

0.2

23.1

3.580

0.174

27.5

13.75

527.0

25.5

---

---

---

3.575

0.173

 

\(\left(={R}_{{{{{{\rm{c}}}}}}}\right)\)

\(\left(={\lambda }_{{{{{{\rm{c}}}}}}}\right)\)

      
  1. *The size-dependent three distinct linewidths are expressed in units of \({{{{{\rm{nm}}}}}}\), which is in accordance with Eq. (3): \({\varGamma }_{{{{{{\rm{EXP}}}}}}}\left(\lambda \right)\equiv {\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)+{\varGamma }_{{{{{{\rm{QC}}}}}}}\left(\lambda \right)+{\varGamma }_{{{{{{\rm{SD}}}}}}}^{{{{{{\rm{ext}}}}}}}(\lambda )\), where \({\prime} \lambda {\prime}\) inside each parenthesis emphasizes that the corresponding PL spectrum is plotted as a function of the PL wavelength (\(\lambda\)).
  2. *\({\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}\,}\left(\lambda \right)\) in Table 2 designates an extracted \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left(\lambda \right)\) value which is obtained using Eq. (3) and the following three contributions: (i) the size-dependent experimental \({\varGamma }_{{{{{{\rm{EXP}}}}}}}\left(\lambda \right)\) value obtained from the corresponding PL-\(\lambda\) spectrum, (ii) the theoretically computed \({\varGamma }_{{{{{{\rm{QC}}}}}}}\left(\lambda \right)\) value, and (iii) the experimentally extracted \({\varGamma }_{{{{{{\rm{SD}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)\) value. We can readily obtain the following relation between \({\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)\) and \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left(R\right),\) where \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left(R\right)\) denotes the size-dependent linewidth solely caused by the exciton-LO phonon coupling (\({\varGamma }_{{{{{{\rm{LO}}}}}}}\)): \({\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)={\varGamma }_{{{{{{\rm{LO}}}}}}}\left(R\right)+({\varGamma }_{{{{{{\rm{o}}}}}}}-{\varDelta }_{R})\) (Methods for details).
  3. *We semi-empirically found that \({\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)={\varGamma }_{{{{{{\rm{LO}}}}}}}\left(R\right)\) because \({\Gamma }_{{{{{{\rm{o}}}}}}}={\Delta }_{{{{{{\rm{R}}}}}}}\) for \(R\le 4.95\ {{{{{\rm{nm}}}}}}\) and \({\varGamma }_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\lambda \right)={\varGamma }_{{{{{{\rm{LO}}}}}}}\left({bg}\right)+({\varGamma }_{{{{{{\rm{o}}}}}}}-{\varDelta }_{{{{{{\rm{R}}}}}}})\) for \(R\ge 4.95\ {{{{{\rm{nm}}}}}}\) where \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left({bg}\right)\) denotes the \({\varGamma }_{{{{{{\rm{LO}}}}}}}\) value that corresponds to the bulk grains (\(18.3\ {{{{{\rm{nm}}}}}}\) for MAPbBr3). \({\varGamma }_{{LO}}\) for sufficiently coarsened grains in a polycrystalline film probably meets this requirement that \({\varGamma }_{{{{{{\rm{LO}}}}}}}={\varGamma }_{{{{{{\rm{LO}}}}}}}\left({bg}\right)\).
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