Author Correction: The elphbolt ab initio solver for the coupled electron-phonon Boltzmann transport equations

Correction to: npj Computational Materials https://doi.org/10.1038/s41524-022-00710-0, published online 07 February 2022

The original version of this article contained typographical errors in Equation (6) and Equation (7).

The correct expression for Eq. (6) is

$$\begin{array}{l}g_{{{{\mathrm{kq}}}}}^{smn} = \sqrt {\frac{\hbar }{{2m_\tau \hbar \omega _{s{{{\mathrm{q}}}}}}}} \frac{1}{{N_{{{\mathrm{e}}}}N_{{{{\mathrm{ph}}}}}}}\mathop {\sum}\limits_{{{{\mathbf{R}}}}_{{{\mathrm{e}}}}{{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}} {\exp \left( {i{{{\mathbf{k}}}} \cdot {{{\mathbf{R}}}}_{{{\mathrm{e}}}} + i{{{\mathbf{q}}}} \cdot {{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}} \right)} \cr \\ \qquad\quad\,\,\,\,\times \mathop {\sum}\limits_{s^\prime m^\prime n^\prime } {g_{{{{\mathbf{R}}}}_{{{\mathrm{e}}}}{{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}}^{s^\prime m^\prime n^\prime }} U_{nn^\prime {{{\mathrm{k}}}}^\prime }U_{m^\prime m{{{\mathrm{k}}}}}^{\dagger} u_{s^\prime s{{{\mathrm{q}}}}} + g_{{{{\mathrm{kq}}}}}^{{{{\mathrm{Fr}}}},smn}. \end{array}$$

which replaces the previous incorrect version:

$$\begin{array}{l} g_{{{{\mathbf{kq}}}}}^{smn} = \sqrt {\frac{\hbar }{{2m_\tau \hbar \omega _{s{{{\mathbf{q}}}}}}}} \frac{1}{{N_{{{\mathrm{e}}}}N_{{{{\mathrm{ph}}}}}}}\mathop {\sum}\limits_{{{{\mathbf{R}}}}_{{{\mathrm{e}}}}{{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}} {\exp \left( {i{{{\mathbf{k}}}} \cdot {{{\mathbf{R}}}}_{{{\mathrm{e}}}} + i{{{\mathbf{q}}}} \cdot {{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}} \right)} \\ \qquad\quad\,\,\times \,g_{{{{\mathbf{R}}}}_{{{\mathrm{e}}}}{{{\mathbf{R}}}}_{{{{\mathrm{ph}}}}}}^{smn}U_{n{\mathbf{k}}^\prime }^ \ast U_{m{{{\mathbf{k}}}}}u_{s{{{\mathbf{q}}}}} + g_{{{{\mathbf{kq}}}}}^{{{{\mathrm{Fr}}}},smn},\\ \end{array}$$

(6)

The correct expression for Eq. (7) is

$$\begin{array}{l}g_{{{{\mathrm{kq}}}}}^{{{{\mathrm{Fr}}}},smn} = i\frac{{e^2}}{{\varepsilon _0V}}\mathop {\sum}\limits_\tau {\sqrt {\frac{\hbar }{{2m_\tau \hbar \omega _{s{{{\mathbf{q}}}}}}}} \left[ {U_{{\mathrm{k}}^\prime }U_{{{\mathrm{k}}}}^{\dagger} } \right]_{nm}} \\ \mathop {\sum}\limits_{{{{\mathcal{G}}}} \ne - {{{\mathrm{q}}}}} {\frac{{\left( {{{{\mathrm{q}}}} + {{{\mathcal{G}}}}} \right) \cdot Z_\tau ^ \ast \cdot \xi _{\tau ,s{{{\mathrm{q}}}}}}}{{\left( {{{{\mathrm{q}}}} + {{{\mathcal{G}}}}} \right) \cdot \epsilon ^\infty \cdot \left( {{{{\mathrm{q}}}} + {{{\mathcal{G}}}}} \right)}} \times \exp \left[ { - i\left( {{{{\mathrm{q}}}} + {{{\mathcal{G}}}}} \right) \cdot {{{\mathrm{r}}}}_\tau } \right]} .\end{array}$$

which replaces the previous incorrect version:

$$\begin{array}{l}g_{{{{\mathbf{kq}}}}}^{{{{\mathrm{Fr}}}},smn} = i\frac{{e^2}}{{\varepsilon _0V}}\mathop {\sum}\limits_\tau {\sqrt {\frac{\hbar }{{2m_\tau \hbar \omega _{s{{{\mathbf{q}}}}}}}} U_{n{{{\mathbf{k}}}}^\prime }^ \ast U_{m{\mathbf{k}}}} \mathop{\sum}\limits_{{\mathcal{G}} \ne-{\mathbf{q}}} \frac{\left( {{\mathbf{q}} + {\mathcal{G}}} \right) \cdot Z_\tau^\ast \cdot {\mathbf{\xi}}_{\tau ,s{\mathbf{q}}}} {{\left( {{\mathbf{q}} + {\mathcal{G}}} \right) \cdot \epsilon ^\infty \cdot \left( {{\mathbf{q}} + {\mathcal{G}}} \right)}}\\\,\qquad\qquad\times\, \left\langle {n{\mathbf{k}}^\prime} \right|\exp \left[ { - i\left( {{\mathbf{q}} + {\mathcal{G}}} \right) \cdot {\mathbf{r}}_\tau } \right] \left|m{\mathbf{k}} \right\rangle,\end{array}$$

(7)