contribution.

![(Color online) The conductivity of monolayer borophane as a function of the chemical potential $\mu$, in the presence of long-range impurity potential along the $x$ ($\sigma_{xx}$), and $y$ ($\sigma_{yy}$), directions. The conductivity of the graphene is also shown.[]{data-label="fig3"}](fig2){width="9.5cm"}

![(Color online) The electrical conductivity of borophane along the $x$ ($\sigma_{xx}$), and $y$ ($\sigma_{yy}$), directions as a function of chemical potential $\mu$, in the presence of short-range electro-magnetic scatterers for $\alpha=0.5$. The conductivity of graphene is also shown.[]{data-label="fig4"}](fig3){width="8.5cm"}

Electro-magnetic scatterers
---------------------------

In this section, we expand our theoretical analysis to a scattering by magnetic impurities seated on the surface of borophane. Magnetic scattering in dilute charged magnetic impurities, containing short-range electric and ferromagnetically ordered magnetic potentials, whose magnetic moments are along the $i$-th direction, is described using the operator $\hat{V}=V_0 (\alpha + \sigma_i) $ [@Rushforth:2007_a; @Rushforth:2007_b], where the dimensionless quantity $\alpha$ is the ratio of the electric and magnetic parts of the impurity potential for an electro-magnetic scatterer. Considering scattering off a $\delta$-scatterer potential of $\hat{V}$, the matrix elements of $\hat{V}/V_0$ in the basis (\[wave\]) can be calculated as follows

$$\begin{aligned}
\vert\langle {\vec{k}}'|\hat{V}/V_0^x|{\vec{k}}\rangle\big\vert &=&
  \frac{1}{2}\big[\alpha(1+e^{i(\beta_{k'}-\beta_k)})+e^{i\beta_{k'}} + e^{-i\beta_k}\big],\\
\vert\langle {\vec{k}}'|\hat{V}/V_0^y|{\vec{k}}\rangle\big\vert &=&
  \frac{1}{2}\big[\alpha+e^{-i\beta_k}(i+\alpha e^{i\beta_{k'}})-i\cos\beta_{k'}+\sin\beta_k \big],\\
\vert\langle {\vec{k}}'|\hat{V}/V_0^z|{\vec{k}}\rangle\big\vert &=&
  \frac{1}{2}\big[(1+\alpha)e^{i(\beta_{k'}-\beta_k)}+\alpha-1\big],\\\end{aligned}$$

It is straightforward to 