Czarnetzki Uwe
Institute for Plasma and Atomic Physics, Ruhr University Bochum, 44780 Bochum, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Dec;88(6):063101. doi: 10.1103/PhysRevE.88.063101. Epub 2013 Dec 3.
A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e.g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e.g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.
基于电子前沿的阶跃分布、恒定截面的电荷交换碰撞、鞘层内可忽略的电离以及可忽略的离子动力学等假设，开发了一种用于电容性放电中平面射频（rf）鞘层的简单分析模型。连续性方程、动量守恒方程和泊松方程被合并为一个关于离子漂移速度平方的单一积分 - 微分方程，即所谓的鞘层方程。从动力学玻尔兹曼方程出发，特别关注动量平衡近似流体方程的推导及其有效性。鞘层方程中的积分出现在屏蔽函数中，该函数考虑了电子密度时间平均值对鞘层空间电荷的相对贡献。结果表明，屏蔽函数对有效鞘层参数的变化相当不敏感。定义解的两个参数分别是最大鞘层扩展与离子平均自由程以及德拜长度的比值。引入了屏蔽函数的一个简单通用解析表达式。借助该表达式，获得了无碰撞以及高碰撞情况下的近似解析解，这些解与精确数值解比较吻合。一个简单的过渡公式允许应用于所有碰撞程度。此外，这些解用于计算鞘层的所有静态和动态量，例如离子密度、场和电流。进一步推导了无碰撞和碰撞情况下的射频Child - Langmuir定律。该模型的一个重要部分是鞘层电压波形的先验知识。这种波形是基于单个鞘层的三次电荷 - 电压关系推导出来的，同时考虑了具有任意对称性的放电中的两个鞘层以及自洽自偏压。假设外部施加的射频电压为正弦波，尽管该模型可以扩展到任意波形，例如用于双频放电。该模型明确计算了高度不对称放电情况下电荷 - 电压关系中的三次校正参数。结果表明，三次校正在一般情况下适中，但在无碰撞情况下更明显。将分析结果与文献中的实验数据进行了比较，这些实验数据是通过对各种气体和压力下电容性鞘层中的平均场和动态场进行激光电场测量获得的。整体上发现吻合得非常好。
