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When I was reading section 3.3, Density of States, in the theses ‘Electronic Phenomena in 2D Dirac-like Systems: Silicene and Topological Insulator Surface States’, I encountered a problem. I don’t know how to derive from (1) to (4). $$N(\omega)=\sum_k \delta\left(\omega-E(\boldsymbol{k})\right)\tag{1}$$ Where $$E\left(\boldsymbol{k}\right)= \pm \sqrt{\hbar^2 v^2 k^2+\Delta^2}\tag{2}.$$ The author used this transformed formula $$\sum_k \rightarrow \frac{1}{(2 \pi)^2} \int d^2 k\tag{3}$$ And finally got $$N(\omega)=\frac{|\omega|}{2 \pi \hbar^2 v^2} \Theta\left(|\omega|-\left|\Delta\right|\right).\tag{4}$$

Qmechanic
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HRTCWQ
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1 Answers1

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$$\begin{align}N(\omega)~\stackrel{(1)+(3)}{=}&\int_{\mathbb{R}^2}\frac{d^2k}{(2\pi)^2}\sum_{\pm}\delta(\omega\!\mp\!E({\bf k}))\cr ~=~&\int_{\mathbb{R}_+}\frac{kdk}{2\pi} 2|\omega|\delta(\omega^2\!-\!E({\bf k})^2)\cr ~\stackrel{(2)}{=}~&\frac{|\omega|}{2\pi} \int_{\mathbb{R}_+}\!d(k^2)~ \delta(\omega^2\!-\!\hbar^2 v^2 k^2\!-\!\Delta^2)\cr ~=~&\frac{|\omega|}{2\pi\hbar^2 v^2} \Theta(\omega^2\!-\!\Delta^2). \end{align}$$

Qmechanic
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