Table 1 Extracted structural distortion parameters of 2D lead bromide perovskites with different cyclic organic cations

From: Regulating Pb off-centering distortion for white-light emission in 2D halide perovskites

 

Pb off-center displacement parameter \({\sigma }_{2}^{2}\) (deg2)

Octahedral angle variance \({\sigma }_{1}^{2}\) (deg2)

Bond length quadratic elongation \(\left\langle \lambda \right\rangle\)

Out-plane tilting distortion \({D}_{{{\rm{out}}}}\) (deg)

In-plane tilting distortion \({D}_{{{\rm{in}}}}\) (deg)

Tilting distortion \({D}_{{{\rm{tilt}}}}\) (deg)

C3

1.6 × 10−8

4.0329

1.0000012

22.27

31.17

33.82

C4

1.3 × 10−8

8.74011

1.0000248

21.89

19.30

31.73

C5

8.0 × 10−6

25.4142

1.0000873

1.14

26.30

26.54

C6

174.9

34.73785

1.0006711

0

21.14

21.13

  1. The octahedral distortion parameters can be evaluated from the octahedral angle variance \({\sigma }_{1}^{2}=\frac{1}{11}{\sum }_{i=1}^{12}{\left({\alpha }_{i}-90\right)}^{2}\), the Pb off-center displacement parameter \({\sigma }_{2}^{2}=\frac{1}{2}{\sum }_{i=1}^{3}{\left({\beta }_{i}-180\right)}^{2}\), the bond length quadratic elongation \(\left\langle \lambda \right\rangle=\frac{1}{6}{\sum }_{i=1}^{6}{\left({d}_{i}/{d}_{0}\right)}^{2}\), where \({\alpha }_{i}\), \({\beta }_{i}\), \({d}_{i}\) and \({d}_{0}\) are the Br‒Pb‒Br bond angle, Br‒Pb‒Br bond angle between non-neighboring Pb‒Br bonds, Pb‒Br bond length, and mean Pb‒Br bond length, respectively. For an ideal octahedron, \({\sigma }_{1}^{2}=0\), \({\sigma }_{2}^{2}=0\) and \(\left\langle \lambda \right\rangle=1\). Note that the \({\sigma }_{1}^{2}\) is not related with \({\sigma }_{2}^{2}\) given that \({\sigma }_{2}^{2}\ne 0\) indicates local inversion symmetry breaking and thus long pair active. The out-plane tilting distortion is extracted from Br‒Pb‒Br bond angle and it is described in detail in Supplementary Note 1.
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