Electronic stopping power in a narrow band gap semiconductor from first principles

The direction and impact parameter dependence of electronic stopping power, along with its velocity threshold behavior, is investigated in a prototypical small-band-gap semiconductor. We calculate the electronic stopping power of H in Ge, a semiconductor with relatively low packing density, using time-evolving time-dependent density-functional theory. The calculations are carried out in channeling conditions with different impact parameters and in different crystal directions for projectile velocities ranging from 0.05 to 0.6 atomic units. The satisfactory comparison with available experiments supports the results and conclusions beyond experimental reach. The calculated electronic stopping power is found to differ in different crystal directions; however, strong impact parameter dependence is observed only in one of these directions. The distinct velocity threshold observed in experiments is well reproduced, and its nontrivial relation with the band gap follows a perturbation theory argument surprisingly well. This simple model is also successful in explaining why different density functionals give the same threshold even with substantially different band gaps.

Figure 1

The total energy of the electronic subsystem as a function of the projectile displacement is shown by the dotted (black) line (for a projectile traveling along the $\left[011\right]$ direction of Ge at a velocity of $0.6$ a.u.). The solid (red) line shows the adiabatic total energy of the electronic subsystem along the same trajectory. The dashed (blue) line shows the difference between the two, i.e., the nonadiabatic energy contribution.

Figure 2

Ge supercell in the [001] direction with H in a channel.

Figure 3

Electronic stopping power $\left(S_e\right)$ vs velocity $\left(v\right)$ of a H projectile in bulk Ge along different crystal directions as obtained from TD-DFT and compared with the experimental measurements (empty triangle dat points) reported in Ref. [35]. The trajectories in all the three directions are along the centers of respective channels with one additional trajectory in the [011] direction (empty square data points) at a very low impact parameter $(0.24$ Bohr position 1 in Fig. 10).

Figure 4

Schematic illustration of the relationship between an indirect band gap and the threshold velocity. The arrow shows a common tangent line from the top valence band to the bottom of conduction band.

Figure 5

Band structure of bulk Ge, calculated using PBE (dashed blue line) and LDA (solid black line). The valence band maxima from the two calculations are aligned with each other for clarity. The two solid (red) arrows illustrate the threshold velocity corresponding to electron-hole excitations in both cases following Eq. (2) in the [001] direction. The dotted (magenta) arrow shows the same (LDA only) in the [111] direction.

Figure 6

The ESP, calculated using the PBE (dashed blue line with triangle data points) and LDA (solid black line with circle data points) functionals, in the [001] direction. The dashed (red) line shows the threshold velocity estimated from the band structure.

Figure 7

The projected electronic densities along the trajectories of projectile in different channels, top [001], middle [111], bottom [011]. The depicted planes are defined by the projectile direction of propagation $\left(z\right)$ and a high symmetry perpendicular direction $d$ (the $\left[011\right]$ in case of the $\left[001\right]$ channel). The electron density increases from dark to bright.

Figure 8

The projected density is averaged over the $z$ axis for all three channels.

Figure 9

Electronic energy against distance along the different projectile trajectories in the [001] direction. The projectile velocity for all the trajectories is $0.5$ a.u.. The inset shows a sectional view of the [001] channel and the trajectories. The gray circles represent Ge atoms in different transverse planes (defining the channel), while the black circles show the projectile positions for different impact parameters.

Figure 10

Electronic energy against distance along the different projectile trajectories in the [011] direction. The projectile velocity for all the trajectories is $0.5$ a.u.. The inset shows a sectional view of the [011] channel and the trajectories. The gray circles represent Ge atoms in different transverse planes (defining the channel), while the black circles show the projectile positions for different impact parameters.

Figure 11

The solid and dashed lines represent abinit and siesta calculations, respectively.