Ballistic correction to the density of states in an interacting three-dimensional metal

We study the tunneling density of states (DOS) in an interacting disordered three-dimensional metal and calculate its energy dependence in the quasiballistic regime, for the deviation from the Fermi energy $E-E_F$, exceeding the elastic scattering rate. In this region, the DOS correction originates from the interplay of the interaction and single-impurity scattering. Depending on the distance between the interaction point and the impurity, one should distinguish (i) the smallest scales of the order of the Fermi wavelength and (ii) larger spatial scales of the order of $\hslash v_F/|E-E_F|$, where $v_F$ is the Fermi velocity. In two dimensions, the large-scale contribution prevails, resulting in a nearly universal DOS correction. The peculiarity of Friedel oscillations in three dimensions is that the contributions from small and large scales are typically comparable, making the DOS correction sensitive to the details of the interaction and demonstrating a significant particle-hole asymmetry. On the other hand, we show that the nonanalytic part of the DOS is determined by large scales and can be expressed in terms of the Fermi-surface characteristics only.

Figure 1

Energy dependence of the quasiballistic correction (3) to the tunneling DOS for the Yukawa interaction (19) at different screening wave vectors $\kappa /p_F=0.2$, 2, $\infty$ (pointlike interaction). Dashed lines are a sketch of the diffusive square-root contribution (1) in the vicinity of the Fermi energy.

Figure 2

(a) Fock (exchange) and (b) Hartree diagrams for the average DOS (measured at point ${\mathit{r}}_0$) of a disordered interacting metal in the ballistic energy window $|E-E_F|\gg 1/\tau$. Here solid, zigzag, and dashed lines represent electron Green function, electron-electron interaction, and disorder potential correlation function, respectively.

Figure 3

Energy dependence of the quasiballistic DOS correction (3) for the pointlike interaction ($\kappa /p_F=\infty$) in the presence of the quartic unharmonicity of the spectrum [Eq. (16)] with $\alpha p_F=0$ (parabolic dispersion), 0.25, 0.4.

Figure 4

Energy dependence of the Fock, Hartree, and total contributions to $\delta {\nu }_{\text{ball}}\left(E\right)$ [see Eq. (3)] for the Yukawa interaction with the screening parameter $\kappa =p_F$.