Towards an integrated modeling of the plasma-solid interface

doi:10.1007/s11705-019-1793-4

Front. Chem. Sci. Eng.

2019, Vol. 13

Issue (2) : 201-237 https://doi.org/10.1007/s11705-019-1793-4

REVIEW ARTICLE

Towards an integrated modeling of the plasma-solid interface

Michael Bonitz¹(

), Alexey Filinov^1,^2,³, Jan-Willem Abraham¹, Karsten Balzer⁴, Hanno Kählert¹, Eckhard Pehlke¹, Franz X. Bronold⁵, Matthias Pamperin⁵, Markus Becker², Dettlef Loffhagen², Holger Fehske⁵

¹. Institute for Theoretical Physics and Astrophysics, Kiel University, 24098 Kiel, Germany
². Leibniz Institute for Plasma Research (INP), D-17489 Greifswald, Germany
³. Joint Institute for High Temperatures RAS, 125412 Moscow, Russia
⁴. Computing Center of Kiel University, D-24118 Kiel, Germany
⁵. Institute of Physics, Greifswald University, D-17489 Greifswald, Germany

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Abstract

Solids facing a plasma are a common situation in many astrophysical systems and laboratory setups. Moreover, many plasma technology applications rely on the control of the plasma-surface interaction, i.e., of the particle, momentum and energy fluxes across the plasma-solid interface. However, presently often a fundamental understanding of them is missing, so most technological applications are being developed via trial and error. The reason is that the physical processes at the interface of a low-temperature plasma and a solid are extremely complex, involving a large number of elementary processes in the plasma, in the solid as well as fluxes across the interface. An accurate theoretical treatment of these processes is very difficult due to the vastly different system properties on both sides of the interface: Quantum versus classical behavior of electrons in the solid and plasma, respectively; as well as the dramatically differing electron densities, length and time scales. Moreover, often the system is far from equilibrium. In the majority of plasma simulations surface processes are either neglected or treated via phenomenological parameters such as sticking coefficients, sputter rates or secondary electron emission coefficients. However, those parameters are known only in some cases and with very limited accuracy. Similarly, while surface physics simulations have often studied the impact of single ions or neutrals, so far, the influence of a plasma medium and correlations between successive impacts have not been taken into account. Such an approach, necessarily neglects the mutual influences between plasma and solid surface and cannot have predictive power.

In this paper we discuss in some detail the physical processes of the plasma-solid interface which brings us to the necessity of coupled plasma-solid simulations. We briefly summarize relevant theoretical methods from solid state and surface physics that are suitable to contribute to such an approach and identify four methods. The first are mesoscopic simulations such as kinetic Monte Carlo and molecular dynamics that are able to treat complex processes on large scales but neglect electronic effects. The second are quantum kinetic methods based on the quantum Boltzmann equation that give access to a more accurate treatment of surface processes using simplifying models for the solid. The third approach are ab initio simulations of surface process that are based on density functional theory (DFT) and time-dependent DFT. The fourths are nonequilibrium Green functions that able to treat correlation effects in the material and at the interface. The price for the increased quality is a dramatic increase of computational effort and a restriction to short time and length scales. We conclude that, presently, none of the four methods is capable of providing a complete picture of the processes at the interface. Instead, each of them provides complementary information, and we discuss possible combinations.

Keywords plasma physics surface science plasma- surface modeling DFT nonequilibrium Green functions

Corresponding Author(s): Michael Bonitz

Online First Date: 06 May 2019 Issue Date: 22 May 2019

Cite this article:

Michael Bonitz,Alexey Filinov,Jan-Willem Abraham, et al. Towards an integrated modeling of the plasma-solid interface[J]. Front. Chem. Sci. Eng., 2019, 13(2): 201-237.

URL:

https://academic.hep.com.cn/fcse/EN/10.1007/s11705-019-1793-4
https://academic.hep.com.cn/fcse/EN/Y2019/V13/I2/201

Fig.1 LTP are one of three main current research topics in plasma physics, aside from magnetic fusion and dense plasma (warm dense matter, WDM) [1]. These systems cover a huge parameter range in the density-temperature plane. LTP (the blue box) range from low (electron) density to atmospheric pressure (right edge). Representatives of solids facing the plasma are metals and semiconductors (electron-hole plasmas) sketched by the red areas. Relevant dimensionless parameters are the classical coupling parameter,

Γ = e 2 / r ¯ k B T

, the quantum coupling parameter,

r s = r ¯ / a B

and the degeneracy parameter of the electrons,

Θ = k B T / E F

, with

r ‾

, a _B and E _F denoting the mean inter-particle distance, the Bohr radius and the Fermi energy, respectively

Fig.2 Low-temperature, low-to atmospheric pressure plasmas being composed of neutrals (atoms, molecules), ions and electrons comprise a number of very unusual properties: They are non-isothermal (T _i ≠ T _e), far from thermal equilibrium (non-Maxwellian velocity distributions), and they contain electrons and ions of a very broad range of kinetic energies. In the plasma boundary region (“sheath”) ions may reach high energies that can be exploited for materials modiﬁcation, sputtering and ion implantation. At the same time, electrons with energies in the eV-range are able to excite and ionize neutrals and trigger chemical reactions

Fig.3 (a) Sketch of the current approach to include surface properties in plasma simulations via phenomenological parameters such as sputter rates R ^sput, electron and ion sticking coefﬁcients, S _e, S _i and the SEE coefﬁcient, g _e. While the plasma is treated by advanced approaches, the atomic structure of the solid and the surface is not resolved. (b) In contrast, in surface science, atomic level information of the surface is taken into account, whereas plasma effects are approximated via independent impacts of neutrals or ions

Fig.4 (a) SEE yield per argon atom or ion for a broad variety of metals (symbols). “Clean metals” refers to beam experiments where the surface was cleaned via ion sputtering. “Dirty metals” denotes measurements following varying degrees of surface exposure to oxygen, to water or ambient gas. Figure from Ref. [12] where additional details are given. (b) Effect of SEE on ion density in the bulk of a AC discharge (13.56 MHz, p = 5 Pa, for varying electrode voltage), for different SEE models: No SEE (black) to a full treatment of ion and atom induced SEE (green). From Ref. [11], figure are reprinted with permission of the authors

Fig.5 Sketch of the plasma-solid interface which comprises the plasma sheath and the plasma facing activated layers of the solid

FT-2

. Among the relevant processes are diffusion, adsorption (“sticking”) and desorption of neutrals, penetration (stopping) of ions and electron transfer between solid and plasma. The inﬂuence of the plasma on the solid and vice versa is a major challenge for a predictive theoretical treatment and require a combination of various theoretical approaches, see Fig. 7

Fig.6 Sketch of the physical processes at the plasma-solid interface–from the largest to the smallest length scale. (a) The electric double layer (on the scale of the Debye length, on the plasma side, and a few nanometers, in the solid) resulting from electron depletion in the plasma sheath (cf. Fig. 3) is characterized by the local difference of the nonequilibrium ion and electron densities and is accompanied by electron accumulation in the solid which is inﬂuenced by the processes in ﬁgure parts (b) and (c). (b) On the scale of the surface roughness (typically nanometers) the surface exhibits local variations of the morphology and chemical composition. (c) Atomic scale modiﬁcation of the surface and the plasma sheath caused by individual particle impacts, charge transfer, chemical reactions, etc. The relevant processes are indicated inside the ﬁgure parts

Fig.7 Theoretical methods for the description of the plasma-solid interface

This concept is based on a research project devoted to plasma-surface physics that is being developed at Kiel University and was ﬁrst presented by M. Bonitz at the conference “Quo vadis—complex plasmas”, Hamburg, August 2015, and at the GEC in Bochum, October 2015

, as sketched in Fig. 5. Some of the processes of interest are listed in the ﬁgure. Note the dramatically different length scales and the very different properties of the plasma and the solid, requiring fundamentally different methods to be applied on the plasma and the solid side. Standard methods for the bulk solid are DFT, BSE, DMFT, and QMC. To simulate surface processes (central box), additional non-adiabatic (time-dependent) approaches are required: MD, KMC, Quantum Boltzmann equation, Born-Oppenheimer MD (BOMD), TDDFT, NEGF and ab initio NEGF (AI-NEGF). To account for the complex interactions between plasma and solid, the corresponding methods have to be properly linked: Plasma simulations should provide the momentum dependent ﬂuxes

J a p

of all species “a” to the surface whereas surface simulations deliver the corresponding ﬂuxes

J a s

that leave the surface. Bulk solid simulations provide the band structure e _l and reactive force ﬁelds (FF), whereas surface simulations return the updated surface morphology, chemical modiﬁcations etc. For details see text

Fig.8 Approximate length and time scales accessible with different simulation methods for plasma-surface applications that are listed in Fig. 7. The shortcuts are the same as in Fig. 7 except for DFT-MD, which is equivalent to BOMD, and QBE which stands for Quantum Boltzmann equation. Note that the comparison is only qualitative as different methods may apply to different processes. Also, the physical time resolution is often much less (larger the minimum scale) than the required time step that is dictated by numerical stability. The ab initio approaches TDDFT and NEGF resolve the electronic length and time scales and apply to ultrafast processes. DFT-MD has a more limited resolution of electronic relaxation processes. The upper limits of DFT, TDDFT and NEGF are set by the required basis dimension and accessible number of time steps. QBE resolves spatial details on the level of 100 inter-atomic distances and the relaxation time of the electrons in the solid. The upper length limit is determined by the imposed accuracy (level of coarse graining, pink arrow). MD propagates only the heavy particles neglecting electronic degrees of freedom. The accessible simulation dimensions can be increased via parallelization. Simulation times are restricted by the number of time steps and can, in some cases, be extended by additional “acceleration” methods (blue arrows). On the plasma side, PIC-MCC simulations resolve approximately one tenth of the electron Debye length and one plasma period and may extend to centimeters and milliseconds. Fluid simulations contain an additional coarse graining with respect to the particle velocities which limits their lower scale limits, compared to PIC, but extends their upper limits. KMC may, in principle, extend plasma-surface simulations to minutes and millimeters but is not considered here due to its largely uncontrolled character for the present applications, see text. Figure adapted from Ref. [28]

Fig.9 (Color online) Illustration of the main ingredients of an Anderson-News model based quantum-kinetic analysis of the neutralization of a helium ion on an aluminum surface characterized by a step potential with depth E _F + F, where E _F is the Fermi energy and F the work function of aluminum. The collision trajectory leading to time-dependent matrix elements enforcing a quantum-kinetic analysis is shown in (a) and the reaction channels included in the modeling are indicated in the (on scale) energy diagram (b) and the reaction diagram (c). For simplicity, the projectile levels shown are the ones far away from the surface, level shifts and broadening due to the interaction with the surface are not visualized. As indicated in (c), there are three routes to the projectile ground state, each one leading to the emission of an electron. The helium ion may capture an electron from the metal by a single-electron transfer (SET), changing its conﬁguration from He⁺ (1s, 1²S_1/2) to either He^* (1s2s, 2³S₁) or He^* (1s2s, 2¹S₀), which may either Auger de-excite to He⁰ (1s², 1¹S₀) or attract another electron from the metal to form a He^*- (1s2s², 2²S_1/2) ion releasing an electron then by auto-detachment (AuD). In addition to these two routes the He⁰ (1s², 1¹S₀) conﬁguration may be also reached by Auger neutralization (AN) of the He⁺ (1s, 1²S_1/2) ion setting also free an electron. Panel (d) ﬁnally depicts the three-level system which can be employed to represent the helium conﬁgurations taking part in the collision. Depending on the process the levels act as ionization or afﬁnity levels. Auxiliary bosons are used to assign the functionality needed to the levels. Not included in the three level modeling, since it is unaffected by the collision, is the spin-up electron in the 1s-shell common to all conﬁgurations listed in (d). For details see Ref. [68].

Fig.10 (Color online) (a) Instantaneous probabilities n ₊(t), n _-(t), n _↑(t), n _↓(t), and n _g(t) for the projectile to be at time t in the He⁺(1s, 1²S_1/2), He^* ^-(1s2s², 2²S_1/2), He^* (1s2s, 2³S₁), He^*(1s2s, 2¹S₀), and the He⁰ (1s², 1¹S₀) conﬁguration together with the instantaneous probability g _e(t) for emitting an electron. The projectile hits the aluminum surface as a positive ion (solid black line) with a kinetic energy E _kin = 60 eV and an angle of incidence with respect to the surface of j = 90^°. The thin vertical line denotes the turning point z _TP = 2.27, separating the incoming (left) form the outgoing (right) branch of the collision trajectory. The ﬁnal probabilities, after the collision is completed, which are also the numbers relevant for plasma modeling, are the values at z = 40 a _B on the outgoing branch. (b) Energy spectrum

γ e (ε q →)

of the emitted electron after the collision is completed

Fig.11 Energy loss of doubly charged helium ions in a graphene sheet [15]. (a) Simulation results for U/J = 1.6, J = 3.15 eV and g = 0.55 and for different cluster size L are compared with SRIM and TDDFT simulations of Ojanperä et al. [149] for an inﬁnite system. The NEGF data (lines: Hartree approximation, red symbols: Second Born self-energy) show good agreement and, in addition, extend to lower projectile energies. (b) A honeycomb cluster with L= 54 sites, U/J = 4, J = 2.8 eV, g = 0, is studied with different self-energy approximations: Second Born, third-order and T-matrix approximation showing a clear impact of correlations

Fig.12 Time-dependent response of a strongly correlated finite honeycomb cluster (L = 12 sites, U/J = 10) to a charged projectile penetrating through the center (point C in (a)). (a) Site averaged double occupation,

d av (t)

1 L ∑ i d i (t)

, for charge Z = 1 (dashed lines), and Z = 2 (full lines). (b) The densities on sites A (full line) and B (dashes) closest to the projectile, for the case Z = 2, after Ref. [206]

Tab.1 Selection of important plasma-surface processes, main surface science methods (as listed in Fig. 7) as well as their quality and limitations ^a)

Fig.13 (Color online) Illustration of the electric response of a ﬂoating dielectric plasma-solid interface. On the right is shown the traditional modeling treating the solid as a black box, characterized by surface parameters such as the electron sticking coefﬁcient and the ion wall recombination probability. The left depicts the processes actually taking place inside the solid when plasma is destroyed. Electrons (e^– ) and ions (i⁺ ), generated by impact ionization in the plasma, hit the dielectric solid, thereby injecting conduction band electrons

(e * −)

and valence band holes (h⁺ ), which after relaxation may either recombine non-radioactively or radioactively. As a result, electron depletion occurs in front of the surface leading to a positive space charge which in turn is balanced by a negative space charge inside the solid (electric double layer). The self-consistent kinetic modeling [212] treats impact ionization in the plasma and recombination in the solid on an equal footing by tracing the am-bipolar charge transport across the interface, allowing for quantum-mechanical reﬂection/transmission and charge transport/relaxation on both sides of the interface

Fig.14 Notation used for the description of an electric double layer at a ﬂoating dielectric plasma-solid interface with negative space charge inside the solid and positive space charge in front of it. Shown are the edges of the conduction (U _*) and valence (U _vb) bands, the edge for the motion of valence band holes (U _h), the potential energies for electrons (U _e) and ions (U _i) on the plasma side, and the energetic range, speciﬁed by the ion’s ionization energy I and its broadening G, in which hole injection occurs due to the neutralization of ions at the interface. Source, reservoir, and quasi-neutral regions are indicated as they will arise in the course of the calculation (adopted from Ref. [212])

Fig.15 Upper panel: Edges for the conduction and valence bands (solid red and blue lines), the free hole motion (dashed blue lines), and the potential energies for electrons and ions on the plasma side (also indicated by solid red and blue lines) for an intrinsic TiO₂ surface in contact with an hydrogen plasma. Shaded regions indicate respectively the reservoir and source which have been set up to provide the correct physical boundary conditions for the double layer. Lower panel: Density proﬁles r _w(z) = [n _*(z) ? n _h(z)]q(?z) and r _p(z) = [n _i(z) ? n _e(z)]q(z). By deﬁnition they are both positive outside the shaded regions, that is, in the regions which are physically relevant. The interface is collisionless on both sides and perfectly absorbing. Material and plasma parameters are c = 4.8 eV, E _g = 3.3 eV,

m e *

= m _e, m _h = 0.8 m _e, e = 6, I = 13.6 eV, G = 2 eV, k _B T _? = k _B T _h = 0.2 eV, k _B T _e = 10 k _B T _i = 2 eV. Due to the somewhat unrealistic temperatures of the charges on both sides of the interface (required to stabilize the numerics) the Debye screening length on the solid side

λ D w = 2.2 × 10 − 6

cm is comparable to the Debye screening length on the plasma side

λ D p = 1.6 × 10 − 6

cm. Due to the absence of collisions the numerical values, however, should not be taken literally. A collisional theory would produce different density and potential proﬁles

Fig.16 Example calculation illustrating the embedding scheme An initially half-filled tight-binding chain (20 sites, nearest-neighbor hopping J, inverse temperature b _s = 100 J ^- ¹) is coupled via a time dependent parameter

γ (t)

γ 0 e − (t − t 0) 2 / (2 τ 2)

to an external energy level e _i = + J giving rise to the transfer of charge. The initial occupation of the energy level is given by n _s = 0.269 (corresponding to an inverse temperature of b _i = 1 J ^- ¹) and

n ‾ σ

1 − n σ

. Furthermore, N _s(t) = S _i n _i _s(t) denotes the total density on the chain

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