Asymmetric radio-frequency capacitively coupled plasma discharges: revisit for better understanding via kinetic simulation and statistical analysis

Abstract The study of geometrically asymmetric Argon discharges aims to develop a precise empirical formula for sheath potential in relation to the electrode area. Analytical models indicate an inverse correlation between the sheath potential and electrode area with varying exponents ( V sh ∝ A − α ). Specifically, the matrix sheath, Child collisionless sheath, and Child collisional-sheath models predicted exponents ( α ) of 2, 4, and 2.5, respectively. Particle-in-cell (PIC) simulations are performed across different geometries, with driven frequencies of 13.56 and 27.12 MHz and pressures of 1 Pa, 5 Pa, and 25 Pa. The average exponent ( α ) from 42 simulation cases is about 0.55 when the two sheaths have the same density. Based on the square-error statistical analysis and considering the plasma density variation at the sheath edge, the relationship is expressed as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> . The higher-energy peak in the ion energy distribution function is linearly related to the gap size, and the charge density decreases as the gap size increases. When the gap size is constant, plasma density rises with increasing plasma pressure and driving frequency. The bimodal ion energy distribution shifts to lower ion energies with higher plasma pressures. The angle of incidence for maximum ion flux increases with rising plasma pressure, allowing ions to strike the powered electrode at a non-normal angle.