Floquet theory for the electronic stopping of projectiles in solids

A general 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 symmetries in a space-time diagonal determined by a projectile with constant velocity moving along a trajectory with crystalline periodicity. This allows for the characterisation of (stroboscopically) stationary solutions, by means of Floquet theory for time-periodic systems. Previous perturbative and non-linear jellium models are recovered from this general theory. An analysis of the threshold velocity effect in insulators is presented based on Floquet quasi-energy conservation.

A general 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 symmetries in a space-time diagonal determined by a projectile with constant velocity moving along a trajectory with crystalline periodicity.This allows for the characterisation of (stroboscopically) stationary solutions, by means of Floquet theory for time-periodic systems.Previous perturbative and non-linear jellium models are recovered from this general theory.An analysis of the threshold velocity effect in insulators is presented based on Floquet quasi-energy conservation.
Particles of radiation shooting through matter interact with the constituent nuclei and electrons and lose their kinetic energy to them.The energy loss per unit length to the electrons (nuclei) is called electronic (nuclear) stopping power S e(n) .It is of great applied interest in various contexts, since materials that can withstand ionic radiation have important applications for medical, nuclear and aerospace engineering industries.From the fundamental side it represents a paradigmatic problem in the context of strongly non-equilibrium electronic systems.Electronic stopping of ions in solids has been studied for over a century.The most popular theoretical paradigm after the early works [1][2][3][4][5] is due to Lindhard in his linearresponse theory of electronic stopping, of general applicability for any host material [6,7], and accessible to first-principles theory [8].It assumes, however, a weak effective interaction between the projectile and the solid, which may be justified at very high velocities [4], but not in general.
A fully non-linear theory for slow projectiles, with velocity v much smaller than the Fermi velocity of the electrons, was proposed for the homogeneous electron liquid (jellium) in the 70's by Ferrel and Ritchie [9] and then developed into a method for calculations by Enchenique, Nieminen and Ritchie [10].It is based on the mapping of the electronic stopping power into the problem of electronic scattering by an impurity in the homogeneous electron liquid, when changing reference frame to the one where the projectile is stationary.A generalisation to arbitrary v was developed later [11][12][13].Although a very successful theory and paradigmatic reference for simple metals, its extension to semiconductors, insulators, transition metals, etc. is very qualitative and limited.
Those are by no means the only theories beyond linear response (see refs.within [37,38]).However, there appears to be no clear physical theory beyond the linearresponse and jellium approximations that can account for the general properties of stopping for arbitrary crystalline systems, an issue that the novel framework presented in this letter intends to address.x'/a t/τ −2 1 x/a FIG.1: Evolution of the sum of a crystalline and a projectile potential in one dimension, both in the laboratory reference frame (left) and in the projectile's (right).a is the lattice parameter, τ = a/v, where v is the projectile's velocity (slope of dotted line).The curves depict potential versus x, and are shifted for different times.Thicker lines indicate times separated by τ .
Model -The theory for jellium [9][10][11][12][13] is implicitly built on the fact that the problem of a projectile of constant velocity v = vv moving in a homogeneous electron liquid, although a time-dependent, non-conservative problem, retains a continuous symmetry and related conservation, which neither stems from time nor space homogeneity, but rather from invariance along a space-time diagonal.The change to the projectile's reference frame aligns this trajectory with the time axis and the problem becomes energy conservative, while still dissipative arXiv:1908.09783v1[cond-mat.mtrl-sci]26 Aug 2019 in the laboratory frame.Consider the same projectile in a crystalline periodic solid, with a spatial periodicity a along its trajectory.The translational invariance becomes discrete along the same line of space-time: the system is invariant under combined space-time translations T * : r → r + nav, t → t + nτ with n integer, and τ = a/|v|.Changing to the projectile's reference frame the problem becomes purely time-periodic with period τ , switching from T * -invariance to T : t → t + τ invariance.This implies that, not only the host electrons, but also the crystalline potential moves past the projectile with velocity −v, which spoils the mapping used in refs.[9][10][11].This is illustrated in Fig. 1, and is the main point exploited in this work, as it allows a treatment based on Floquet theory for time-periodic Hamiltonians [39,40].
A projectile with kinetic energy in e.g. the MeV scale will slow down while exciting the electron system at a rate of a few eV/ Å.It is a strong, non-perturbative excitation, but the slowing down of the projectile is barely noticeable over a significant distance in the atomic scale.As done in the non-linear theory for jellium [9][10][11][12][13], we consider the ideal non-conservative case of a projectile moving at a constant velocity along a rectilinear trajectory in a solid.The following formalism will be limited to non-interacting particles, and the projectile will be represented by a local scalar potential.The method can be straightforwardly generalized to more realistic situations using time-dependent mean-field or Kohn-Sham methods to include realistic crystals, projectiles, and electronelectron interactions.
A general time-dependent lattice-projectile model Hamiltonian, with the T * symmetry in the laboratory frame in which the lattice is at rest, can be written as where H 0 (r ) is the lattice Hamiltonian and V P (r , t) = V P (r − vt) is the potential that describes a projectile with velocity v.In the reference frame moving with the projectile, r = r − vt (primed/unprimed indices indicating lab/projectile reference frame, respectively) the Hamiltonian becomes which is time-periodic with period τ = a/|v|.Floquet theory.-According to the Floquet theorem [39,40], for a time periodic Hamiltonian there are timedependent solutions to the Schrödinger equation of the form where φ α (r, t), the Floquet mode, has the same time periodicity of the Hamiltonian.The real parameter ε α , the Floquet quasi-energy, uniquely defines solutions up to multiples of ω, ω = 2π/τ , and it is a conserved quantity.
A first important consequence of this theory is that the Floquet modes in the projectile's frame define the stationary solutions to the stopping problem in the lab frame.In previous theoretical work [7,10,11], stationary solutions were either assumed or a direct consequence of key approximations.Their existence and character appear now naturally from Floquet theorem.Stationary now means T -periodic, or stroboscopic, i.e. timeindependent if looking at it at instants t = t 0 + nτ for n ∈ Z.It does not mean these are the only expected solutions.In addition to transients related to occasional perturbations, one can also foresee deviations like the flapping instability recently proposed [33], which represents the analog of a charge density wave along the T *symmetric direction in space-time, a generalization of the time-crystal idea.
Stopping from Bloch-Floquet scattering theory.-Thestopping problem in the lab frame becomes a scattering one for the Floquet modes in the projectile's, in analogy with the theory for jellium, replacing energy conservation by quasi-energy conservation and treating time t as just an additional degree of freedom at the same level of a spatial coordinate [41,42].The asymptotic scattering states away from the projectile consist of the Bloch states of the crystal transformed to the projectile's frame.The Floquet Hamiltonian for the latticeprojectile system is defined as H(r, t) = H 0 (r, t) + V (r), where H 0 (r, t) = H 0 (r + vt) − i ∂ ∂t is the lattice Floquet Hamiltonian, periodic with period τ , whose complete set {φ α (r, t)} of eigenmodes are readily extracted from the Bloch states in the lab frame ψ nk (r ) = e ik•r u nk (r ), with energy E n (k) and band index n, which, transformed to the projectile's frame, become [43] ψ n,k (r, t) = u nk (r + vt)e i(k−mv/ )•r e −iεn(k)t/ , (5) where and m is the electron mass.By comparing with Eq. 3, the quasi-enegies and Floquet modes are immediately identified as ε α = ε n (k) and φ α (ξ) = φ nk (r, t) ≡ u nk (r + vt)e i(k−mv/ )•r (Bloch-Floquet modes henceforth).The Floquet BZ for quasi-energy can be chosen to coincide with the BZ for the Bloch vectors: shifting k by pG 0 (for p ∈ Z and G 0 = (2π/a)v) shifts the quasi-energy by p ω.
Consider an initial Bloch state in the moving frame ψ nki (r, t) = e −iεn(ki)t/ φ nki (r, t): adding the projectile, a perturbation of arbitrary strength which does not break the symmetry T , the periodic mode of the full solution Ψ (±) nki = e −iεn(ki)t/ Φ (±) nki (r, t) with quasi-energy ε n (k i ) can be expressed as an integral equation in the Lippmann-Schwinger spirit, with ξ = (r, t), where the (±) sign indicates outgoing/incoming boundary conditions, and an averaging over one cycle time t is implied.G ± 0 (ε n (k i )|ξ, ξ ) is the propagator for H 0 , which using completeness of the Floquet modes is [41,44] Let us start from a general 1D system for simplicity, where ξ = (x, t).The asymptotic behaviour of Eq. 6 in terms of the incoming and outgoing Bloch-Floquet modes and the scattering amplitudes is The band index m and momenta k f of the scattered states are determined by the quasi-energy conservation condition ε n (k i ) = ε m (k f ), which has in general multiple solutions -see Fig. 2 for an example.It can also be expressed, the explicit expression for the quasi-energy from Eq. 5 (and using 3D notation) as This expression, appearing naturally from quasi-energy conservation, coincides with what obtained from energy and momentum conservation in a collision of an electron with a projectile of mass M P → ∞ [24], and in perturbation theory [45].The possible values of the k's in Eq. 9 are not limited to the 1st BZ, but must be considered in the extended zone scheme as in Fig. 2, or, equivalently, on bands shifted by a multiple of ω.
The scattering state needs to fulfill outgoing boundary conditions, i.e. the group velocity defined as has to point away from the projectile.The state in Eq. 8 differs from the 1D free-particle case, whose asymptotic scattering states are plane waves [10][11][12] with only two outgoing channels per particle (reflected and transmitted) from energy conservation, similarly to Bloch-wave scattering by defects and impurities [46][47][48].From Eq. 6, the scattering amplitudes in Eq. 8 are from conservation of probability flux.From this relation an expression for the energy transfer rate (ETR) to the electron system in the lab frame can be derived.For the single-particle scattering state of Eq. 8, considering the energy flux difference between the outgoing and incoming states one gets where ρ i corresponds the density of the incoming state.The corresponding expression in 3D is where ∆E mn, It is important to note here that the ETR is averaged over a period τ , as the scattering amplitude coefficients are proportional to the the (time-averaged) matrix element V P .Electronic stopping can be defined as S e = Ė/v, where Ė is here the total ETR, to be calculated by considering the contributions of all of the possible transitions between occupied and unoccupied states.At temperature T = 0, assuming occupied bands n and unoccupied bands m and integrating separately over initial and final momenta, it is where the 3D version of Eq. 10 was used.To extend this to T = 0 and partially filled bands the relevant occupation numbers need to be introduced.
FIG. 3: Model one-dimensional insulator with parabolic bands and an indirect band gap (above).Red lines delimit range of possible electron-hole pair transitions compatible with Eq. 9 for projectile velocity v defining their slope.Below: Joint density of states (JDOS) ρ(ω, v) versus excitation energy ω plotted for velocities v 2 > v 1 > v th for the same model.The spectral limits for v 2 show the van Hove singularities related to the red lines above (outer tails due to artificial broadening).
These are very general expressions, which can be related to previous results and theoretical models.The homogeneous electron liquid theory [9][10][11][12][13] is recovered from Eq. 13 for the ETR and from Eq. 14 for the stopping power.Alternatively, if the projectile is treated as a small perturbation, the equivalent of a 1 st Born approximation for Floquet scattering can be used [49]: substituting |Φ (+) nki by |φ nki and assuming a smooth projectile V P (r), the scattering amplitude matrix elements in Eq. 10 become where ∆k = |k f − k i |, and ṼP (∆k) indicates the Fourier transform of V P (r), thereby recovering perturbation theory results (see e.g.45).
Threshold velocity for insulators.-Thelow-v limit for the stopping of ions in a gapped material is now analyzed, which is not well described by earlier theories, and proved to be quite controversial in experiments [35,36,50].We do it for a model insulators with parabolic energy bands around the gap and isotropic effective masses m e and m h for electrons and holes.Let us consider an indirect band gap E g with the bottom of the conduction band displaced by k 0 from the top of the valence band, and a projectile travelling with velocity v parallel to k 0 .A joint density of states (JDOS) can be defined in analogy with optical transitions [51], which offers interesting insights (see Fig. 3).Stopping power S e (v) follows by suitable integration in energy of the JDOS.
For the parabolic model in Fig. 3 no stopping is allowed below a threshold velocity v th .For an actual insulator, however, the threshold behavior is less clean.In fact, the adiabatic limit v → 0 is quite non-trivial, as illustrated in Fig. 4: by quasi-energy conservation (Eq.9) transitions are allowed for arbitrarily small v even for gapped solids.This is shown in the figure using the repeated zone scheme implied in Eq. 9, where the lines of allowed transitions decrease in slope with decreasing v. Importantly, this picture is general and independent of perturbation theory [45,52].The S e (v) curve is characterized by a series of onset velocities, or partial thresholds, v (p) th , for p ∈ Z ≥0 (slopes of red lines and red dots in the upper and lower panels of Fig. 4, respectively), defined by In the low-v limit (large p), v th can be now approximated as summing over all replicas l beyond the p-th, where γ l , relating to the scattering rate for transitions to the l-th replica, is taken to decay with l, and assumed constant for states within the l-th parabola, and where f th ) is the stopping power contribution of the l-th replica.Close to each replica onset, f (v) ∝ v m , where m depends on dimension (m = 1 in 1D, m = 2 in 3D).Assuming an algebraic decay, γ l ∼ l −µ , S e (v In summary, the presented Floquet theory provides a natural framework for the description of the stroboscopic stationary states arising in electronic stopping processes, as well as the reference states for possible instabilities along the space-time symmetric direction, analogous to CDWs in space, or time crystals in time.A general expression for the electronic stopping power has been derived and analysed.Previous perturbative [45,49] and non-linear jellium [10][11][12] theories are recovered, in the limits of either weak coupling or homogeneous electron host.Floquet quasi-energy conservation has allowed the characterization of velocity thresholds in insulators, which prove to be far from trivial.The theory provides a paradigm for the understanding of electronic stopping processes, and should lead to predictive computational schemes possibly more efficient than today's.

FIG. 4 :
FIG. 4: Top: Partial threshold velocities (slopes of red lines, v p th ) for the different replicas of parabolic bands in the extended zone scheme, corresponding to shifted Floquet modes, as in Fig. 2. Bottom: effective threshold behaviour for S e versus v in the small v limit, for a 3D indirect gap model with γ l ∝ e −αl λ .Red dots correspond to first values of v (p) th , slopes of red arrows in top panel.v 0 is the threshold velocity for transitions within the 1st BZ.Inset: log S e vs 1/v.