Linear response of doped graphene sheets to vector potentials

A two-dimensional gas of massless Dirac fermions (MDFs) is a very useful model to describe low-energy electrons in monolayer graphene. Because the MDF current operator is directly proportional to the (sublattice) pseudospin operator, the MDF current-current response function, which describes the response to a vector potential, happens to coincide with the pseudospin-pseudospin response function. In this work, we present analytical results for the wave vector- and frequency-dependent longitudinal and transverse pseudospin-pseudospin response functions of noninteracting MDFs. The transverse response in the static limit is then used to calculate the noninteracting orbital magnetic susceptibility. These results are a starting point for the construction of approximate pseudospin-pseudospin response functions that would take into account electron-electron interactions (for example at the random-phase-approximation level). They also constitute a very useful input for future applications of current-density-functional theory to graphene sheets subjected to time and spatially varying vector potentials.

Figure 1

Different regions for the behavior of ${\chi }_{{\sigma }_x{\sigma }_x}^{\left(0\right)}\left(\mathit{q},\omega \right)$. In this figure we have introduced dimensionless variables: $\overline{q}=q/k_{\text{F}}$ and $\overline{\omega }=\omega /{\epsilon }_{\text{F}}$ The regions B.1 and B.2 are characterized by a continuum of interband electron-hole pairs. The regions A.2 and A.3 are characterized by a continuum of intraband electron-hole pairs. $\mathfrak{I}m{\chi }_{{\sigma }_x{\sigma }_x}^{\left(0\right)}\left(\mathit{q},\omega \right)=0$ in regions A.1 and B.3.

Figure 2

The imaginary part of the longitudinal pseudospin response function, $-\mathfrak{I}m{\chi }_{{\sigma }^x{\sigma }^x}^{\left(0\right)}\left(\mathit{q}=q\widehat{x},\omega \right)$ [in units of the density-of-states at the Fermi level, $\nu \left(0\right)={\epsilon }_{\text{F}}/\left(2\pi v^2\right)$], as a function of $\omega /{\epsilon }_{\text{F}}$. (a) $q=0.5k_{\text{F}}$, (b) $q=1.0k_{\text{F}}$, (c) $q=2.0k_{\text{F}}$, and (d) $q=3.0k_{\text{F}}$. Singularities at $\omega =vq$ are clearly visible.[18]

Figure 3

The imaginary part of the transverse pseudospin response function, $-\mathfrak{I}m{\chi }_{{\sigma }^x{\sigma }^x}^{\left(0\right)}\left(\mathit{q}=q\widehat{y},\omega \right)$ [in units of $\nu \left(0\right)$], as a function of $\omega /{\epsilon }_{\text{F}}$. (a) $q=0.5k_{\text{F}}$, (b) $q=1.0k_{\text{F}}$, (c) $q=2.0k_{\text{F}}$, and (d) $q=3.0k_{\text{F}}$.

Figure 4

The real part of the longitudinal pseudospin response function, $-\mathfrak{R}e{\chi }_{{\sigma }^x{\sigma }^x}^{\left(0\right)}\left(\mathit{q}=q\widehat{x},\omega \right)$ [in units of $\nu \left(0\right)$], as a function of $\omega /{\epsilon }_{\text{F}}$. (a) $q=0.5k_{\text{F}}$, (b) $q=1.0k_{\text{F}}$, (c) $q=2.0k_{\text{F}}$, and (d) $q=3.0k_{\text{F}}$. The filled circles in panels (a) and (b) mean that the function $-\mathfrak{R}e{\chi }_{{\sigma }^x{\sigma }^x}^{\left(0\right)}\left(\mathit{q}=q\widehat{x},\omega \right)$ takes exactly the value 1 [in units of $\nu \left(0\right)$] at $\omega =vq$.

Figure 5

The real part of the transverse pseudospin response function, $-\mathfrak{R}e{\chi }_{{\sigma }^x{\sigma }^x}^{\left(0\right)}\left(\mathit{q}=q\widehat{y},\omega \right)$ [in units of $\nu \left(0\right)$], as a function of $\omega /{\epsilon }_{\text{F}}$. (a) $q=0.5k_{\text{F}}$, (b) $q=1.0k_{\text{F}}$, (c) $q=2.0k_{\text{F}}$, and (d) $q=3.0k_{\text{F}}$.