In this section we will consider a perfect Si (diamond) crystal with a single atom oscillating around its lattice site while all of the other atoms remain fixed. The oscillating atom will have a z position given by
z(t) = z0+ A sin(ωt), (7.4)
where A is the amplitude of the oscillation, z0is the equilibrium position and ω is the driving frequency. An atom oscillating around its lattice site will not alter the electronic structure of the bulk material if the amplitude of the oscillation is small, allowing us to study the electronic stopping power of Si with no elevator state.
The work of Mason et al. [40] considered a single atom oscillating in a Cu-like metal. From the simulation data and perturbation theory they constructed a molecular dynamics (MD) force field to describe the directional dependence of the electronic stopping power in a face- centered cubic metal. In this section we will use the techniques developed by Mason et al. to examine the electronic stopping power for a single atom oscillating in the z direction in Si.
In MD simulations the electronic stopping power due to an atom moving with a velocity v is often represented by a drag force (F) of the form [37]
F(t) = −βv, (7.5)
where β is the electronic damping. The electronic energy transfer (Ee) at a time t is given
by [40]
Ee(t) = −
ˆ tn
t0
Time-Dependent Tight Binding Results 89 and substituting Eq. (7.5) into the above yields
Ee(t) =
ˆ tn
t0
dt β(t) |v(t)|2, (7.7)
where we have allowed the electronic damping to be time dependent. We are interested in the energy transfer due to an oscillating ion (Eq. (7.4)) during the mth period, but Mason et al. [40] showed that the electronic damping is independent of the number of oscillations and can be treated as a constant, hence Eq. (7.7) becomes
Ep =
ˆ (m+1)τ
mτ
dt β|A|2ω2cos2(ωt) = 1
2β|A|2ω2τ, (7.8)
where we have introduced τ = 2π
ω and Ep is the electronic energy transfer for a single period.
The total electronic energy transfer for m complete oscillations is given by
Epm= 12β|A|2ω2mτ, (7.9)
and mτ can be interpreted as the time elapsed during m complete periods. Hence, by differentiating with respect to time and rearranging
β = 2 |A|2ω2
dEe
dt , (7.10)
where Ep
(mτ) = E∆te ≈ dEdte is measured over an integer number of periods.
7.4.1 Perturbation Theory
For an atom oscillating in a perfect crystal with an angular frequency ω, the probability (P) of a transition between a pair of states separated by an energy gap of & within a time t is given by [40]
P(&, t) = |V |
2πt
2! {δ(& − !ω) + δ(& + !ω)} , (7.11) where V is the matrix element between the states and is assumed to be a constant. The electronic energy transfer between the pair of states is given by
∆Ee(&, t) = &P(&, t). (7.12)
Eq. (7.12) cannot be applied to our simulation results in its current form because it does not take into account the density of states (DOS) of the Si crystal. Including the valence (Dv) and conduction (Dc) band DOS in Eq. (7.12) gives
∆Ee(t) =
ˆ
vdx
ˆ
cdy (y − x)Dv(x) Dc(y) P(y − x, t), (7.13) where the integrals are over all of the states in the valence and conduction bands, and we have assumed that the valence band is initially occupied and the conduction band is initially unoccupied. By applying a change of variables from the conduction band energy (y) to the energy difference (& = y − x), Eq. (7.13) simplifies to
∆Ee(t) =
ˆ
vdx
ˆ Ec−x
Eg−x
−15 −10 −5 0 5 10 Energy (eV) 0 2 4 6 8 10 12 14 DOS (eV − 1)
Figure 7.6: The DOS for 6144 atoms with Γ point sampling and a Gaussian smearing of 0.02 eV. The DOS was calculated using the Kwon TB model in spICED.
where Eg and Ec are the bottom and top of the conduction band respectively and we have
chosen for the top of the valence band to be at zero eV. The total electronic energy transfer is obtained by substitution of Eq. (7.11) into Eq. (7.14) and produces the result
∆Ee(t) = |
V|2πωt 2
ˆ
vdx Dv(x)Dc(x + !ω), (7.15)
where we have used the delta functions to evaluate the integral over &. The electronic damping (Eq. (7.10)) is therefore given by
β = |V|
2π
|A|2ω ˆ
vdx Dv(x)Dc(x + !ω) (7.16)
allowing for a direct comparison to our simulation results. We use the DOS calculated by spICED for 6144 atoms, with Γ point sampling and a Gaussian smearing of 0.02 eV as shown by Fig. 7.6.
7.4.2 Results
In this subsection we examine the results for a perfect crystal of Si with a single atom oscillating around its lattice site with a frequency ω and amplitude A = 0.1 Å in the z direction. The results were calculated using the Kwon [60] TB model for systems containing 4096 and 6144 atoms. The analysis uses the GEM (see Sec. 7.1). The perturbation theory expression Eq. (7.16) was fitted to the simulation data by adjusting the matrix element. A value of V = 1.509 eV produces exceptional agreement to the simulation data, as shown by Fig. 7.7. We find that the simulation results for the two different cell sizes are almost identical, demonstrating that the 4096 atom simulation cell is converged with respect to the system size.
Time-Dependent Tight Binding Results 91 0 5 10 15 20 25 ¯hω(eV) 0.0 0.5 1.0 1.5 β (eV fs ˚ A − 2)
Perturbation theory spICED: 4096 atoms spICED: 6144 atoms
Figure 7.7: The electronic damping due to an atom oscillating in a perfect crystal for simulations using 4096 and 60144 atoms. The perturbation theory is given by Eq. (7.16) where the DOS is from Fig. 7.6 and V = 1.509 eV.
The data in Fig. 7.7 shows a clear threshold frequency in the electronic drag. The threshold frequency is approximately equal to 0.74 eV, which is the band gap of the Kwon [60] TB model. The hard threshold appears contradictory to the channelling results (see Sec. 7.3, chapters 4 and 9), however this is not the case. A perfect crystal has extended states, which are time-independent. The addition of a channelling ion creates a time-dependent elevator state within the band gap, which allows for low energy excitations by the directly transporting electrons across the gap (see Ch. 4) and by harmonic excitations (see Ch. 9 for details). Let us consider a perfect crystal with a single atom that oscillates with a small amplitude (fractions of an Å) then, to first order, there will be no significant change to the perfect crystal energy eigenvalues, hence all of the states remain time-independent. If we then allowed for the oscillating atom to have a large amplitude, then we would expect the electronic eigenvalues to depend on the atom’s position, hence we would create a time- dependent state and allow for pre-threshold excitations.
The maximum in the electronic damping, as shown by Fig. 7.7, is an artificial feature due to the finite size of the Kwon [60] conduction band DOS. The maximum in the electronic damping corresponds to an excitation of about seven eV and is equal to the conduction band width, see Fig. 7.6. For excitations of energy greater than seven eV there are no empty states in the conduction band for electrons from the top of the valence band to occupy. Thus, there are fewer electrons excited. As the frequency of the oscillator increases more valence band electrons want to be excited to energies higher than the top of the conduction band DOS, resulting in the steady decrease in electronic damping. The largest possible excitation is from the bottom of the valence band (−15 eV) to the top of the conduction band (five eV), hence, as shown by Fig. 7.7, the electronic damping reduces to zero when !Ω ! 20 eV.