Table 2 Three distinct contributions to the size-dependent experimental linewidth of MAPbBr3 CNCs, expressed in \({{{{{\bf{meV}}}}}}\) value.

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

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

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

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

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

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{EXP}}}}}}}\left(\hslash {{{{{\boldsymbol{\omega}}}}}} \right)\) \(\left({{{{{\rm{meV}}}}}}\right)\)

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{SD}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\hslash {{{{{\boldsymbol{\omega}}}}}} \right)\) \(\left({{{{{\rm{meV}}}}}}\right)\)

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{QC}}}}}}}\left(\hslash {{{{{\boldsymbol{\omega}}}}}} \right)\) \(\left({{{{{\rm{meV}}}}}}\right)\)

\({{{{{\boldsymbol{\Gamma }}}}}}_{{{{{{\rm{LO}}}}}}}^{{{{{{\rm{ext}}}}}}}\left(\hslash {{{{{\boldsymbol{\omega}}}}}} \right)\) \(\left({{{{{\rm{meV}}}}}}\right)\)

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

2.7

1.35

475

3.966

197

192

10.4

—

0.293

3.2

1.6

498

3.783

169

160

8.5

0

0.256

5.0

2.5

514

3.665

136

87

5.2

44

0.206

6.0

3.0

518

3.637

122

67

4.2

51

0.185

7.5

3.75

522

3.609

101

27

3.6

70

0.154

8.7

4.35

523

3.602

100

24

3.2

73

0.154

9.9

4.95

524

3.595

106

20

2.7

83

0.161

19.5

9.75

525.1

3.587

105

10

1.4

94

0.161

22.5

11.25

526.0

3.582

111

10

1.3

100

0.168

23.7

11.85

526.2

3.580

114

10

0.9

103

0.174

27.5

13.75

527.0

3.575

114

—

—

—

0.173

 

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

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

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

     
  1. * The size-dependent three distinct linewidths are expressed in units of energy (\(\hslash \omega\)) in Table 2. In contrast, the three linewidths are expressed in \({nm}\) unit in Table 1.
  2. * The following equations can interconvert the FWHMs between the PL-\(\omega\) and PL-\(\lambda\) spectra: \({\varGamma }_{{{{{{\rm{SD}}}}}}}\left(\omega \right)={\varGamma }_{{{{{{\rm{SD}}}}}}}\left({{\hslash }}\omega \right)/{{\hslash }}=2{{{{{\rm{\pi }}}}}}c{\varGamma }_{{{{{{\rm{SD}}}}}}}\left(1/\lambda \right)=2{{{{{\rm{\pi }}}}}}c\{\,\frac{1}{\left({\lambda }_{{{{{{\rm{p}}}}}}}-\tfrac{1}{2}{\varGamma }_{{{{{{\rm{SD}}}}}}}(\lambda )\right)}-\frac{1}{\left({\lambda }_{{{{{{\rm{p}}}}}}}+\tfrac{1}{2}{\varGamma }_{{{{{{\rm{SD}}}}}}}\left(\lambda \right)\right)}\,\}=\frac{2{{{{{\rm{\pi }}}}}}c\,{\varGamma }_{{SD}}(\lambda )}{\left\{{\lambda }_{{{{{{\rm{p}}}}}}}^{2}-{\left(\tfrac{1}{2}{\varGamma }_{{{{{{\rm{SD}}}}}}}(\lambda )\right)}^{2}\right\}}\approx \frac{2{{{{{\rm{\pi }}}}}}c\,{\varGamma }_{{{{{{\rm{SD}}}}}}}\left(\lambda \right)}{{\lambda }_{{{{{{\rm{p}}}}}}}^{2}},\) where \({\lambda }_{{{{{{\rm{p}}}}}}}^{2}\gg {\left(\tfrac{1}{2}{\varGamma }_{{{{{{\rm{SD}}}}}}}\left(\lambda \right)\right)}^{2}.\) Thus, we establish the following relation: \({\varGamma }_{{{{{{\rm{SD}}}}}}}\left({{\hslash }}\omega \right)={hc}\,{\varGamma }_{{{{{{\rm{SD}}}}}}}(\lambda )/{\lambda }_{{{{{{\rm{p}}}}}}}^{2},\) where \({\varGamma }_{{{{{{\rm{SD}}}}}}}(\lambda )\) represents the contribution of the CNC-size distribution to the net linewidth in the PL-\(\lambda\) spectrum (expressed in \({nm}\) unit). Similarly, \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left({{\hslash }}\omega \right)\) and \({\varGamma }_{{{{{{\rm{QC}}}}}}}\left({{\hslash }}\omega \right)\) are estimated using the following relations: \({\varGamma }_{{{{{{\rm{LO}}}}}}}\left({{\hslash }}\omega \right)={hc}\,{\varGamma }_{{{{{{\rm{LO}}}}}}}(\lambda )/{\lambda }_{{{{{{\rm{p}}}}}}}^{2}\) and \({\varGamma }_{{{{{{\rm{QC}}}}}}}\left({{\hslash }}\omega \right)={hc}\,{\varGamma }_{{{{{{\rm{QC}}}}}}}(\lambda )/{\lambda }_{{{{{{\rm{p}}}}}}}^{2}\).
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