Electron Elevator: Excitations across the Band Gap via a Dynamical Gap State

We use time-dependent density functional theory to study self-irradiated Si. We calculate the electronic stopping power of Si in Si by evaluating the energy transferred to the electrons per unit path length by an ion of kinetic energy from 1 eV to 100 keV moving through the host. Electronic stopping is found to be significant below the threshold velocity normally identified with transitions across the band gap. A structured crossover at low velocity exists in place of a hard threshold. An analysis of the time dependence of the transition rates using coupled linear rate equations enables one of the excitation mechanisms to be clearly identified: a defect state induced in the gap by the moving ion acts like an elevator and carries electrons across the band gap.

Figure 1

Electronic stopping power of a Si self-interstitial moving along the $⟨001⟩$ channel as a function of projectile kinetic energy $K$. $E_{\text{th}}$ and $E_{\text{th}}^{\prime }$ are calculated using Eq. (1) with the LDA band gap and an energy difference of 0.2 eV, respectively. There are three distinct regions: in the metallike regime ($K>3000\mathrm{eV}$, $v>1.45\AA {\mathrm{fs}}^{-1}$) the stopping is linear in projectile velocity, with a slope of $1.47\mathrm{eV}\mathrm{fs}{\AA }^{-2}$; in the band-edge regime ($60<K<3000\mathrm{eV}$, $0.2<v<1.45\AA {\mathrm{fs}}^{-1}$) the stopping rises rapidly with velocity; in the prethreshold regime ($K<60\mathrm{eV}$, $v<0.2\AA {\mathrm{fs}}^{-1}$) the stopping is nonzero. The inset shows the stopping power as a function of ion velocity minus the metallic stopping power [defined as $S_m=\gamma \times (v-v_0)$ for $v>v_0$, where $\gamma$ and $v_0$ are fitted to the high velocity points]. Stopping powers from the Srim model [11] (extrapolated to low velocity) and experiment (lowest velocity point measured in Ref. [27]) are also shown.

Figure 2

Left: evolution of the adiabatic energy eigenvalues at the supercell $\mathrm{\Gamma }$ point as a function of the position of the projectile along the $⟨001⟩$ channel in Si. The addition of the channeling ion creates a midgap “elevator state” (red line), the energy of which oscillates as the projectile moves along the channel. The smallest difference between the elevator energy and the valence (conduction) band edge is approximately (less than) 0.2 eV. Right: evolution of the adiabatic energy eigenvalues during a Born-Oppenheimer quantum molecular dynamics simulation in which a Si interstial in a relaxed crystal of 64 initially stationary Si atoms was given an initial velocity of $0.2\AA {\mathrm{fs}}^{-1}$ in the $⟨001⟩$ direction. All 65 atoms were allowed to move in response to the interatomic forces generated during the simulation. The initial velocity of $0.2\AA {\mathrm{fs}}^{-1}$ is too low for channeling to take place, but the elevatorlike behavior of the defect states remains apparent. Similar elevatorlike behavior was observed when the initial velocity of the projectile was along the $⟨110⟩$ direction.

Figure 3

Evolution of the occupation of the elevator state as an ion of kinetic energy 39 eV ($v\approx 0.164\AA {\mathrm{fs}}^{-1}$) travels down the $⟨001⟩$ channel. The defect state gains electrons when its energy is close to the valence band edge and loses electrons when its energy is close to the conduction band edge; the loss of electrons from the defect state is accompanied by a corresponding increase in the occupation of the conduction band. The elevator state thus controls the excitations of electrons to the conduction band. The dashed lines show a fit to the rate-equation model described in the text.

Figure 4

Bottom: the velocity dependence of the rates of excitation directly from the valence band to the conduction band ($R^{\text{cv}}$), from the valence band to the defect level ($R^{\text{dv}}$), and from the defect level to the conduction band ($R^{\text{cd}}$). Top: the ratio of the rate of direct valence-conduction excitations to the rate of excitations via the defect state.