3.9.2 Spectral Function and Local Density of States

The spectral function is defined as

$$\begin{array}{l}\displaystyle A({\bf r},{\bf r'};E) \ = \ i ... ...- \ G^\mathrm{a}({\bf r},{\bf r'};E) \right] \ . \end{array}$$ (3.82)

The spectral function provides information about the nature of the allowed electronic states, regardless whether they are occupied or not, and can be considered as a generalized density of states. The diagonal elements of the spectral function give the local density of states

$$\begin{array}{l}\displaystyle \rho({\bf r};E) \ = \ \frac{1}... ...left[ G^\mathrm{r}({\bf r},{\bf r};E)\right] \ . \end{array}$$ (3.83)

The trace of the spectral function represents the density of states

$$\begin{array}{l}\displaystyle N(E)\ = \ \mathrm{Tr}\left[A(E... ...] \ = \ \int d{\bf r} \ A({\bf r},{\bf r};E) \ . \end{array}$$ (3.84)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors