Kinetic modeling and simulation of the planar multipole resonance probe

为了控制工艺等离子体，需要原位测量其状态变量，特别是电子密度和电子温度。这个研究项目将有助于这一探索。它侧重于一个特定的诊断方案，行业兼容的诊断方法主动等离子体共振光谱（APRS）。APRS通过记录等离子体对由电探针施加的射频信号的响应来获得关于等离子体的信息。以APRS概念的具体实现为例，即所谓的平面多极共振探头（pMRP），其优点是可以垂直安装在腔室壁中。利用第一性原理揭示了观测到的共振与等离子体内部参数电子密度和电子温度之间的关系。原则上，共振是可以在流体动力学模型内很好地描述的集体效应。这涵盖了观察到的共振频率和电子密度之间的关系。然而，共振的阻尼受到个体效应的强烈影响，个体效应只能在完全动力学模型中进行研究。虽然自1960年代以来就知道这种影响的存在，但尚未完全了解。为了使APRS方案成为一个有用的测量方案，必须在定量的水平上理解动力学效应的影响。实验方法是不可能的，因为负责任的物理发生在探头附近，即，在毫米的空间尺度上，并且仅在高频测量过程本身期间，即，在纳秒的时间尺度上。这实际上是不可能通过实验解决的。因此，本研究将尝试另一种验证方法，即比较两种基于两种完全不同方法的数学上不同的方法。第一个使用功能分析，第二个制定一个积分方程的基础上的动力学方程的线性化。泛函分析方法的优点是碰撞可以被认为是相对容易的，积分方程方法是注定的碰撞自由动态。 为了比较这两种方法，必须将它们都加以扩展，以至少部分地涵盖另一种制度。当这一点实现后，将首次对动力学效应对共振特性的动态影响有一个完整的了解。一旦弄清了探针行为的基本物理方面，将建立一个综合的数学模型，它将把所观察到的共振的频率和阻尼与电子密度和电子温度联系起来。这样的关系将是用于等离子体工艺的监督和控制的期望的改进的评估规则的基础。

To control technological plasmas, the in-situ measurement of their state variables is required, especially of the electron density and the electron temperature. This research project will contribute to this quest. It focuses on a particular diagnostic scheme, the industry-compatible diagnostic method active plasma resonance spectroscopy (APRS). APRS obtains information on a plasma by recording its response to radio frequency signal applied by an electric probe. A specific realization of the APRS concept is taken as example, the so-called planar multipole resonance probe (pMRP) which has the benefit of fitting squarely into the chamber wall. First principles are used to uncover the relation between the observed resonance and the inner plasma parameters electron density and electron temperature. In principle, the resonance is a collective effect which can be described well within a fluid dynamic model. This covers the relation between the observed resonance frequency and the electron density. The damping of the resonance is, however, strongly influenced by individual effects which can be studied only within a fully kinetic model. Although the existence of this influence is known since the 1960s, it is not yet fully understood. This is core of this research project.To turn the APRS scheme into a useful measurement scheme, the influence of the kinetic effects must be understood on a quantitative level. An experimental approach is not possible, as the responsible physics takes place in the vicinity of the probe head, i.e., on the spatial scales of millimeters, and only during the high-frequency measuring process itself, i.e., on the temporal scale of nanoseconds. This is virtually impossible to be resolved experimentally. For that reason, this research will attempt another approach for validation, namely compare two mathematically different methods which are based on two entirely different approaches. The first uses functional analysis; the second formulates an integral equation based on a linearization of the kinetic equation. The functional analytic approach has the advantage that collisions can be considered relatively easily; the integral equation approach is predestined for a collision-free dynamic. To compare the methods, they must both be extended to cover the other regime at least partially. When this is achieved, a complete understanding of the dynamic influence of the kinetic effects on the characteristics of the resonances will be available for the first time.Once the basic physical aspects of probe behavior are clarified, a comprehensive mathe-matical model will be formulated which sets the frequency and the damping of the observed resonance in relation to the electron density and the electron temperature. Such a relation will be the basis for the desired improved evaluation rule to be used for supervision and control of plasma processes.