as $$\begin{aligned}
v_{x}^{\tau}(k)&=&v_{t}+\tau \frac{v_x^{2}k_x}{\sqrt{v_x^{2}k_x^2+v_y^{2}k_2^2}}\\
v_{y}^{\tau}(k)&=&\tau \frac{v_y^{2}k_y}{\sqrt{v_x^{2}k_x^2+v_y^{2}k_2^2}}\end{aligned}$$

The density of states (DOS) can be obtained by solving the following equation

$$\begin{aligned}
\label{eq-54}
 D(\varepsilon) &=& \displaystyle \frac{1}{(2\pi)^2} \int_0^\infty k'\, dk'
  \delta(\varepsilon-\varepsilon_{\vec{k}'})\end{aligned}$$

Performing this integral over energy, one finds the following expansion for the density of states:

$$\begin{aligned}
\label{eq-54}
  D(\varepsilon) =  \frac{k^{\tau}(\varepsilon,\phi) }{v_t\cos(\phi) + \sqrt{v_x^2 \cos^2(\phi)+ v_y^2 \sin^2(\phi)}}\end{aligned}$$

where $\phi=\tan^{-1}(k_y/k_x)$ and the wave vector $ k^{\tau}(\varepsilon,\phi)$ is given by

$$\begin{aligned}
\label{eq-54}
 k^{\tau}(\varepsilon,\phi) =\varepsilon  \frac{-2 v_t\cos(\phi) +\tau \sqrt{2}\sqrt{ v_x^2 +  v_y^2 +  (v_x^2-  v_y^2) \cos(2\phi)}}{v_x^2 + v_y^2 - v_t^2 + (v_x^2 - v_y^2 - v_t^2) \cos(2\phi)}\end{aligned}$$

Interestingly, similar to the low-energy DOS of graphene, DOS of borophane has the well-known linear form, as shown in Fig.\[energyDispersion\](b). In this figure, also the DOS of graphene (dashed-green) is plotted by setting $v_x=v_y=1\times10^{6}\,m/s $ and $v_t=0$.

![(Color online) Schematic drawing of the optimized ground-state structure of borophane lattice, with top (left panel) and side view (right panel). The unit cell is marked with a yellow rectangle, contains two boron (B) atoms and two hydrogen (H) atoms and the basic vectors of the primitive unit cell are indicated by the yellow arrows. The green and white balls represent B and H atoms, respectively. The buckling height is denoted 