Correction to: npj 2D Materials and Applications https://doi.org/10.1038/s41699-020-0141-3, published online 12 May 2020

The authors became aware of a numeric error in a parameter they used (dipolar interactions) in the model to interpret their experimental findings. As a result of this error, the following changes have been made to the original version of this Article:

In “Modelling interactions in multi-exciton complexes” under “Results and discussion”, the forth sentence of the first paragraph originally stated "For values of ℓ ~ 3–4 nm, reference to Fig. 3a shows that Δρ ~ 5–6.5 nm.” In the corrected version “Δρ ~ 5–6.5 nm” is replaced by “Δρ ~ 4.5–5.5 nm”.

The forth sentence after Equation (5) originally read “The experimentally observed blueshift of LIX2 of (8.4 ± 0.6) meV is consistent with a confinement lengthscale in the range ℓ = 2.8–3.3 nm”. In the corrected version “ℓ = 2.8–3.3 nm” is replaced by “ℓ = 2.5–3.0 nm”.

The eighth sentence after Equation (5) originally read “Remarkably, … for a dielectric constant in the range ϵ 3, … and an effective confinement lengthscale of ℓ ≈ 3 nm.” In the corrected version “ϵ 3” is replaced by “ϵ ≈ 3” and “ℓ ≈ 3 nm” is replaced by “ℓ  3 nm”.

The correct version of Fig. 3 is shown below after correcting the dipolar repulsion and exchange splitting energy (~20% differences).

This has now been corrected in both the PDF and HTML versions of the Article.

Besides, “Supplementary Note 11: Calculations of the binding energy of the localized multi-IXs” of the Supplementary Information contained errors in the text and some Supplementary Equations because all dipolar interactions were counted twice.

The original Supplementary Equations (4–7) incorrectly read:

$$U^{2X} = 2 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{2}} \right)^2 + \,2\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(4)
$$U^{3X} = 3 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{{\sqrt 3 }}} \right)^2 + \,6\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(5)
$$U^{4X} = 4 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{{\sqrt 2 }}} \right)^2 + \,8\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3} + \,4\frac{{d^2}}{{4\pi \epsilon }}\left( {\sqrt 2 {{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(6)
$$\begin{array}{ll}U^{5X} = 5 \cdot \displaystyle\frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left({\displaystyle\frac{{\sqrt {50 + 10\sqrt 5 } {{{\mathrm{{\Delta}}}}}\rho }}{{10}}} \right)^2 + \,10\displaystyle\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}\\ \qquad\qquad+ \,10\displaystyle\frac{{d^2}}{{4\pi \epsilon }}\left( {\frac{1}{2}\left( {1 + \sqrt 5 } \right){{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}\end{array}$$
(7)

The correct form of Supplementary Equations (4–7) is:

$$U^{2X} = 2 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{2}} \right)^2 \,+ \,\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(4)
$$U^{3X} = 3 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{{\sqrt 3 }}} \right)^2\, + \,3\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(5)
$$U^{4X} = 4 \cdot \frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\frac{{{{{\mathrm{{\Delta}}}}}\rho }}{{\sqrt 2 }}} \right)^2\, + \,4\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3} \,+ \,2\frac{{d^2}}{{4\pi \epsilon }}\left( {\sqrt 2 {{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}$$
(6)
$$\begin{array}{ll}U^{5X} = 5 \cdot \displaystyle\frac{1}{2}\frac{{\hbar ^2}}{{M\ell ^4}}\left( {\displaystyle\frac{{\sqrt {50 + 10\sqrt 5 } {{{\mathrm{{\Delta}}}}}\rho }}{{10}}} \right)^2\, +\, 5\displaystyle\frac{{d^2}}{{4\pi \epsilon }}\left( {{{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}\\ \qquad\qquad+ \,5\displaystyle\frac{{d^2}}{{4\pi \epsilon }}\left( {\frac{1}{2}\left( {1 + \sqrt 5 } \right){{{\mathrm{{\Delta}}}}}\rho } \right)^{ - 3}\end{array}$$
(7)

The original Supplementary Equations (8–11) incorrectly read:

$$U_0^{2X} = \frac{5}{{12^{3/5}}}\tilde E \approx 1.125\tilde E$$
(8)
$$U_0^{3X} = \frac{{5 \cdot 18^{2/5}}}{6}\tilde E \approx 2.648\tilde E$$
(9)
$$U_0^{4X} = \frac{{5\left( {8 + \sqrt 2 } \right)^{2/5}}}{{2^{2/5}3^{3/5}}}\tilde E \approx 4.806\tilde E$$
(10)
$$U_0^{5X} \approx 7.648\tilde E$$
(11)

The correct form of Supplementary Equations (8–11) is:

$$U_0^{2X} = \frac{5}{{2 \cdot 6^{3/5}}}\tilde E \approx 0.853\tilde E$$
(8)
$$U_0^{3X} = \frac{{5 \cdot 9^{2/5}}}{6}\tilde E \approx 2.007\tilde E$$
(9)
$$U_0^{4X} = \frac{{5\left( {8 + \sqrt 2 } \right)^{2/5}}}{{2^{4/5}3^{3/5}}}\tilde E \approx 3.642\tilde E$$
(10)
$$U_0^{5X} \approx 5.796\tilde E$$
(11)

The original Supplementary Equations (12–15) incorrectly read:

$$E^{2X \to 1X} \approx E^X + 1.125\tilde E$$
(12)
$$E^{3X \to 2X} \approx E^X + 1.523\tilde E$$
(13)
$$E^{4X \to 3X} \approx E^X + 2.158\tilde E$$
(14)
$$E^{5X \to 4X} \approx E^X + 2.842\tilde E$$
(15)

The correct form of Supplementary Equations (12–15) is:

$$E^{2X \to 1X} \approx E^X + 0.853\tilde E$$
(12)
$$E^{3X \to 2X} \approx E^X + 1.154\tilde E$$
(13)
$$E^{4X \to 3X} \approx E^X + 1.635\tilde E$$
(14)
$$E^{5X \to 4X} \approx E^X + 2.154\tilde E$$
(15)

The third sentence after Supplementary Equation (21) originally read “For example, arranging three excitons in a line would result in $$U_{0,{\rm{line}}}^{3{\rm{X}}}$$ ≈ 3.496 $$\tilde E$$ > $$U_0^{3{\mathrm{X}}}$$, a quadexciton with three excitons on a triangle and the fourth one in the center would give $$U_{0,{\rm{triangle}}}^{4{\mathrm{X}}}$$ ≈ 5.493 $$\tilde E$$ > $$U_0^{4{\mathrm{X}}}$$ and a quintexciton consisting of four excitons on a square surrounding the fifth in the trap center would have $$U_{0,{\rm{square}}}^{5{\mathrm{X}}}$$ ≈ 7.845 $$\tilde E$$ > $$U_0^{5{\mathrm{X}}}$$.” In the corrected version “3.496 $$\tilde E$$” is replaced by “2.650 $$\tilde E$$”, “5.493 $$\tilde E$$” is replaced by “4.163 $$\tilde E$$” and “7.845 $$\tilde E$$” is replaced by “5.945 $$\tilde E$$”.

The HTML has been updated to include a corrected version of the Supplementary Information.