Numerical study of bound states for point charges shielded by the response of a homogeneous two-dimensional electron gas

We study numerically the existence and character of bound states for positive and negative point charges shielded by the response of a two-dimensional homogeneous electron gas. The problem is related to many physical situations and has recently arisen in experiments on impurities on metal surfaces with Shockley surface states. Mathematical theorems ascertain a bound state for two-dimensional circularly symmetric potentials $V\left(r\right)$ with ${\int }_0^{\infty }drrV\left(r\right)⩽0$. We find that a shielded potential with ${\int }_0^{\infty }drrV\left(r\right)>0$ may also sustain a bound state. Moreover, on the same footing we study the electron-electron interactions in the two-dimensional electron gas, finding a bound state with an energy minimum for a certain electron gas density.

Figure 1

(Color online) The shielded attractive potentials of Eq. (5) (solid blue curve) and Eq. (6) (dashed red curve) with $\alpha =4$ and $\beta =2\sqrt{2}$, respectively. The dotted black curve refers to a bare Coulomb potential $-1∕r$. The dash-dotted green curve is devoted to the attractive Thomas-Fermi form of Eq. (4).

Figure 2

Energy eigenvalue $E_b^{\left(H\right)}\left(\lambda \right)$ based on Eq. (7) with Eq. (10) for $\Lambda =1$ and $\lambda ∊[0.97,1.03]$.

Figure 3

Binding energy $\mid E_b^{\left(H\right)}\left(\Lambda \right)\mid$ based on Eq. (7) with Eq. (10) for $\lambda =1$ and $\Lambda ∊[1.25,0.75]$.

Figure 4

(Color online) Energy eigenvalues $E_b\left(r_s\right)$ in the shielded field of a negative charge. Data are based on Eq. (7) with $\mu =1∕2$, and $\Lambda =\lambda =1$ in the applied two effective interactions. The solid blue and dashed red curves refer to the hydrogenic [Eq. (5)] and Gaussian [Eq. (6)] models, respectively. See the text for further details.

Figure 5

The radial density $2\pi r{\mid \psi \left(r\right)\mid }^2$ and the potential based on hydrogenic screening of a negative unit charge. These are computed at the $r_s=11.4$ value of the density parameter.

Figure 6

The radial density $2\pi r{\mid \psi \left(r\right)\mid }^2$ and the potential based on Gaussian screening of a negative unit charge. These are computed at the $r_s=17.2$ value of the density parameter.

Figure 7

The coupling-constant $\left(\Lambda \right)$ dependence of the energy eigenvalue for the repulsive case. The results are based on the Gaussian model of Eq. (13) with $\mu =1∕2$ in Eq. (7). The Wigner-Seitz parameter is $r_s=17.2$ and $\lambda =1$.

Figure 8

The shielding-constant $\left(\lambda \right)$ dependence of the energy eigenvalue for the repulsive case. The results are based on the Gaussian model of Eq. (13) with $\mu =1∕2$ in Eq. (7). The Wigner-Seitz parameter is $r_s=17.2$ and $\Lambda =1$.