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).

figure 1

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.