Two-electron bound states near a Coulomb impurity in gapped graphene

We formulate and solve the perhaps simplest two-body bound-state problem for interacting Dirac fermions in two spatial dimensions. A two-body bound state is predicted for gapped graphene monolayers in the presence of weakly repulsive electron-electron interactions and a Coulomb impurity with charge $Ze>0$, where the most interesting case corresponds to $Z=1$. We introduce a variational Chandrasekhar-Dirac spinor wave function and show the existence of at least one bound state. This state leaves clear signatures accessible by scanning tunneling microscopy. One may thereby obtain direct information about the strength of electron-electron interactions in graphene.

Figure 1

Energy functional $E_{\epsilon =-1}(Z_I,Z_O)$ in units of $\mathrm{\Delta }$ for the spin-singlet state as a function of $Z_I$ for $Z=1$, taking $Z_O=0.3$. The green dashed-dotted (red dashed, blue solid) curve is for $\alpha =0.3(0.35,0.4)$, respectively. Horizontal lines indicate the respective threshold energies $\mathrm{\Delta }(1+2\gamma )$. When a minimum exists below the threshold (as is the case for all shown $\alpha$'s), a bound state is present.

Figure 2

Rescaled dimensionless binding energy ${\overline{E}}_{-,\mathrm{b}}$ vs $\alpha$, see Eq. (32), in the spin-singlet sector with $Z=1$.

Figure 3

Optimal values of $Z_I$ (main panel) and $Z_O$ (the inset) vs $\alpha$ for the spin-singlet case with $Z=1$.

Figure 4

Cumulative weight $W_{\lambda }$ vs $\alpha$, see Eq. (35), of the negative energy states in the variational wave function for $Z=1$, where $\lambda =I(\lambda =O)$ corresponds to black circles (red squares).

Figure 5

Radial density $P\left(r\right)$ vs distance $r$ from the impurity (in units of $\hslash v_F/\mathrm{\Delta }$) of the spin-singlet two-body bound state, cf. Eq. (47), for $Z=1$ and several values of $\alpha$. The solid lines correspond to the relativistic case for $\alpha =0.1,0.2$, and 0.3, shown in red, blue, and black colors (from bottom to top), respectively. The dashed lines indicate the corresponding nonrelativistic results.

Figure 6

Radial profile of the pair distribution function $g\left(r\right)$ vs interparticle distance $r$ (in units of $\hslash v_F/\mathrm{\Delta }$) for $Z=1$ and several values of $\alpha$. The solid lines correspond to the relativistic case for $\alpha =0.1,0.2$, and 0.3, shown in red, blue, and black colors (from bottom to top), respectively. The dashed lines indicate the corresponding nonrelativistic results.