Floquet theory for the electronic stopping of projectiles in solids

A theoretical framework for the study of electronic stopping of particle projectiles in crystalline solids is proposed. It neither relies on perturbative or linear-response approximations nor on an ideal metal host. Instead, it exploits the discrete translational invariance in a space-time diagonal for a constant velocity projectile following a trajectory with crystalline periodicity. By means of Floquet theory, (stroboscopically) stationary solutions are characterized, and previous perturbative and jellium models are recovered. The threshold-velocity effect in insulators is analyzed based on quasienergy conservation.

Figure 1

Evolution of crystalline plus projectile potential in (a) the laboratory reference frame and in (b) the projectile. $a$ is the lattice parameter $\tau =a/v$, and $v$ is the projectile velocity. The curves (potential vs $x$) are shifted for different times. Thicker lines indicate times separated by $\tau$.

Figure 2

Bloch-Floquet-mode quasienergies ${\varepsilon }_n\left(\mathbf{k}\right)=E_n\left(\mathbf{k}\right)-\hslash \mathbf{k}·\mathbf{v}+\frac{1}{2}m{v}^2$ of a two-band model along the projectile-periodic direction in reciprocal space, extended zone scheme, and Floquet replicas ${\varepsilon }_{pn}\left(\mathbf{k}\right)$ included (thin lines) $v>0$. The first BZ is highlighted in gray. Quasienergy conserving states are indicated for an incoming mode at $k=0$ (red star). Blue circles and black triangles label allowed scattering modes in the lower and upper bands, respectively, both in the extended zone scheme (on thick curves) and folded back into the first BZ.

Figure 3

(a) Model parabolic bands with indirect band gap in one dimension (1D) for illustration. Red lines delimit possible electron-hole pair transitions compatible with Eq. (10) with projectile velocity $v$ defining their slope. (b) JDOS $\rho (\omega ,v)$ vs excitation energy $\omega$ for velocities $v_2>v_1>v_{\mathrm{th}}$ for the same model. Red vertical lines highlight van Hove singularities.

Figure 4

(a) Partial threshold velocities (slopes of red lines $v_{\mathrm{th}}^{\left(p\right)}$) for replicas of parabolic bands in the extended zone scheme, corresponding to shifted Floquet modes, cf. Fig. 2. (b) Effective threshold behavior for $S_e$ vs $v$ in the small $v$ limit for a 3D indirect gap model with ${\gamma }_l\propto e^{-\alpha l}$. Red dots: $v_{\mathrm{th}}^{\left(p\right)},v_0$ is the threshold velocity for transitions within the first BZ. The inset: $S_e$ (logarithmic scale) vs $1/v$ highlights the quick decay of stopping as $v\to 0$.