Strength of dipolar backflow patterns around slow protons in three- and two-dimensional electron gases

The familiar dipolar backflow in an electron gas around a slowly moving massive impurity represents, in linear response, the averaged induced current far from the impurity and is proportional to the density response function and the forward scattering amplitude within the Born approximation. Here, we calculate the strength of the dipolar density modulation around a slow proton in three- and two-dimensional paramagnetic electron gases, beyond the perturbative linear-response treatment, by using scattering phase shifts at the Fermi energy which satisfy the Friedel-sum rule. These are determined by solving self-consistently the ground-state Kohn-Sham equations for screening. A sign-changing effect, as a function of the electron gas density, is found in the strength in both dimensions. Using the self-consistent phase shifts, a recently proposed expression for the so-called direct charge in three-dimensional electromigration is also investigated.

Figure 1

Strengths of the dipolar backflow around a slow proton $Z=1$ in a $2D$ gas for $r_s∊[0.5,7]$. The leading term, denoted by $h^{\left(1\right)}\left(2D\right)$, and the complete expression, denoted by $h\left(2D\right)$, are based on Eqs. (1, 2) and are represented by open and filled circles, respectively. The density parameter $\left(r_s\right)$ is calculated from the host density as $\pi r_s^2=1∕n_0$.

Figure 2

Strengths of the dipolar backflow around a slow proton $Z=1$ in a $3D$ gas for $r_s∊[0.5,7]$. The leading term, denoted by $h^{\left(1\right)}\left(3D\right)$, and the complete expression, denoted by $h\left(3D\right)$, are based on Eqs. (1, 2) and are represented by open and filled circles, respectively. The open squares, based on Eq. (4), refer to the direct charge $Z_d\left(3D\right)$. The density parameter $\left(r_s\right)$ is calculated from the host density as $(4\pi r_s^3∕3)=1∕n_0$.