3.2.1 Introduction and background
The particle-in-cell (PIC) scheme is a purely kinetic representation of a sys- tem containing ions and electrons, allowing a self-consistent representation of virtually any plasma.
Birdsall(1991) and more recentlyVerboncoeur(2005) have written two com-
prehensive and very interesting reviews on particle-in-cell simulations. Both of them include roots of particle simulations and the basic techniques. Verbon- coeur’s review also treats recent advances made in the field.
duration of the pulse, i.e. 1 ms.
3The B field reported in the earliest papers on the current-free double layer by Charles
and Boswell were wrong by a factor of two; the maximal value of the field was 130 G in the source and not 250 G as shown in figure3.1.
3.2. Particle-in-cell simulations 47
The early days of particle simulations go back to calculations performed
by Buneman (1959) and Dawson (1962) with only a few hundreds of particle
electrons to study instabilities and thermalizing properties, respectively. In these models, the electrical forces due to the space charges were calculated using Coulomb’s law. These models are referred to as particle-particle models, as the interactions between a particle and all the others are taken into account, which is, of course, very computationally expensive. This was improved by the use of particle-mesh models, where the charges are accumulated on a spatial grid on which Poisson’s equation is solved; these models are referred to as cloud-in-cell or particle-in-cell.
The development of modern PIC took place in the late 1960s (Birdsall and
Fuss 1969), 1970s (Langdon and Birdsall 1970) and early 1980s. The earliest
work attempting to include Monte Carlo collisions (electron-neutral collisions) in a PIC simulation was done by Burger (1967) and led to many subsequent works (Vahedi et al. 1993ba,Vahedi and Surendra 1995). Most of the theoreti- cal work on the effects of the spatial grid (Langdon and Birdsall 1970) and the analysis of the time integration (Langdon 1979) was done in the 1970s. Tech- niques used in modeling bounded plasmas have been well described in a number of books and publications (Hockney and Eastwood 1988,Birdsall and Langdon
1985, Vahedi et al. 1993ba). For many years plasma simulations were used to
simulate the plasma bulk, neglecting the sheaths, hence, the use of periodic boundary conditions was very popular. Realistic absorbing boundary condi- tions were first introduced by Birdsall and Langdon (1985), and improved by the use of electron secondary emission (Hagstrum 1956,Harrower 1956,Suren-
dra et al. 1990,Surendra et al. 1990) and by the inclusion of an external circuit
(Verboncoeur et al. 1993) for a more accurate representaion of processing sys-
tems.
The time step and the cell size of full particle-in-cell simulations, where both ions and electrons are treated as particles, have to resolve both the elec- tron plasma frequency and the Debye length, to ensure stable and accurate simulations. These extremely stiff conditions become a real issue when sim- ulating high density plasmas, as the plasma frequency is proportional to the square root of the plasma density. In addition, when dealing with phenomena related to ion transport or whose time scale is large, for example on the order of millisecond, full particle-in-cell simulations become really cumbersome, as they can require several billions of time steps and several days of calculation. Increasing the speed of particle-in-cell simulations is a challenging issue that has been going on for years. As early as in the beginning of the 1980s, Cohen
and Freis(1982) andLangdon et al.(1983) have developed a so-called implicit
scheme for PIC simulations allowing use of larger time steps than with the classical explicit scheme. Accelerating PIC simulation can also be done by the use of adaptive grids and time scales, electron sub-cycling (Adam et al. 1982), consisting of moving the much heavier ions less often than the electrons. More recently, the use of variable macro-particle weight (Coppa et al. 1996, Shon
et al. 2001) was developed. A very interesting review on physical and numeri-
cal technics to accelerate full PIC simulation was written by Kawamura et al.
Interpolation of the force acting on each particle
E→f
Integration of the field equations (Poisson)
ρ→E
Gathering of the charges on the nodes x→ρ Particle pushing (Newton Law) f→v→x Boundary conditions Expansion model Monte Carlo Collisions
Heating process
Classical PIC scheme Additional modules
Figure 3.3: Classical particle-in-cell (PIC) scheme and extra modules.
As other kinetic methods, PIC is very attractive since it provides a self- consistent solution of the fields and particle dynamics using only first principles (Poisson’s equation and Newton’s law) without making any assumptions on the charged particle transport. Particle-in-cell simulations are particularly useful when modeling non-equilibrium plasmas, such as breakdown (Vender et al. 1996) or when modeling plasmas for which the electron transport is not known a priori. The use of PIC simulations generally falls into one of the following two categories. i) PIC can provide insight in areas where the theory is incomplete or inaccurate or when its assumptions cannot be verified experimentally. ii) PIC can be used as an extension to experiment and compared directly to experiment, giving results to clarify or explain the underlying mechanisms involved in some experimental plasmas. This second category is still limited by the complexity of the real plasma chemistry, the more or less complex geometry of the system, the real surface behavior etc., that have to be simplified to make the simulation tractable.
3.2.2 General particle-in-cell scheme
Particle-in-cell is a purely kinetic representation of a system containing ions and electrons, considered as individual particles, moving under the influence of their own self-consistent electric field (Birdsall and Fuss 1969,Langdon and Birdsall
1970,Hockney and Eastwood 1988, Birdsall and Langdon 1985). PIC simula-
tions use the first principles (Poisson’s equation and Newton’s laws) only. Each particle of the simulation is actually a macro-particle allowed to represent a large number of real particles (on the order of 109 or 1010particles per macro-particle for one-dimensional simulations; this number can be decreased even more with the improvement of computational resources) and which can move inside the
3.2. Particle-in-cell simulations 49
simulated domain. With a small number of these macro-particles (typically between 104 and 105 for one-dimensional simulations), a realistic steady-state
plasma can be obtained in a few hours on a modern desktop computer.
Let us describe the general one-dimensional PIC scheme assuming a planar geometry. The simulated region is divided into Nc cells resulting in a grid of
Nc+ 1 nodes. Electric field only is considered (electrostatic simulation), thus each particle is pushed (accelerated and moved) using Newton’s law
mdv
dt =qE, (3.1)
wherem is the mass of the particle,q its charge andvits velocity. The electric fieldE is given by
E=−∂Φ
∂x, (3.2)
and where the potential Φ is integrated from Poisson’s equation
∂2Φ ∂x2 =− ρ ε0 =−e ε0 (ni−ne), (3.3) where ni and ne are the ion and the electron densities, respectively; e is the elementary charge and ε0 the vacuum permittivity. Physical quantities, such
as the potential, the electric field, the position, the velocity. are normalized according to the characteristic quantities of the system, such as the cell size ∆x, the time step ∆t, the elementary chargeeetc. Scaling the physical quantities i) decreases the number of operations, such as multiplications by physical constant ii) and minimizes round-off errors.
The steps of the conventional PIC scheme with additional modules are sum- marized in Figure3.3 and are described below.
Charge assignment and field interpolation
The charge density ρ is assigned to each node of the grid by accumulation of the charges of the various species (electrons and ions). The accumulation of the charges can be done following various models such as Nearest-Grid-Point (NGP), Cloud-in-Cell (CIC) etc. These models are discussed in detail inHock-
ney and Eastwood(1988) andBirdsall and Langdon (1985).
It is generally accepted that the best tradeoff between computational cost and accuracy is the CIC scheme. It is a first-order weighting model involving the two nearest grid points. The charge density on each mesh point is cal- culated by linearly distributing the charge of each macro-particle to its two nearest grid points. Likewise, the electric field at a given position is obtained by interpolating the two nearest field values.
Charge gathering and evaluation of the electric field at the position of a macro-particle have to use the same interpolation kernel (NGP, CIC etc.), as it was shown that using different interpolation kernels leads toself-forces4 (Bird-
sall and Langdon 1985) and higher heating rates (Mardahl and Verboncoeur
1997). 4
Electric potential and electric field on the nodes
The electric potential is calculated by solving Poisson’s equation discretized as follows Φi−1−2Φi+ Φi+1 ∆x2 =− ρi ε0 , (3.4)
where ∆xis the cell length andithe index of the considered node. Equation3.4 leads to a tri-diagonal system of equations, which can be solved by any classical algorithm for tri-diagonal systems (see Press et al. 1992, for example ). The electric field is then obtained by using the following finite-difference equation
Ei =−
Φi+1−Φi−1
2∆x . (3.5)
For the first (resp. last) node, the electric field is a linear extrapolation of the field of the second and third (resp. penultimate and antepenultimate) nodes; various boundary condition models are proposed and discussed byVerboncoeur et al. (1993).
Particle pushing
Particles are accelerated by integrating Newton’s law (equation3.1) discretized as follows vt+1 2 =vt− 1 2 + qEt∆t m . (3.6)
Particles are then moved according to
xt+1 =xt+vt+1
2∆t. (3.7)
Equations3.6and3.7lead to the classical leap-frog scheme where positions and velocities are not known simultaneously.
To allow stable and accurate simulations of cold plasmas, the time step and the size of the cell have to meet the following criteria (Birdsall and Langdon
1985,Hockney and Eastwood 1988)
ωp∆t2, ∆x < λD,
(3.8) where ωp is the electron plasma frequency
ωp =
s
nee2
meε0
, (3.9)
and λD the Debye length
λD = r ε0kBTe e2n e . (3.10)