# Vertex Single Theory

### Overview

The incoming ions have an energy component in the X direction perpendicular to the sampling aperture EI and an energy component in the Y direction EII parallel to the sampling aperture ### Variable Aspect Ratio

The aspect ratio determines if the EII energy component is such that the ion gets collected at the aperture electrodes or passes through the apertures for detection at the collector of the RFEA. ### Variable Aspect Ratio Using a Variable Bias

The electric field applied in the EI direction determines if the EII energy component is such that the ion gets collected at the aperture electrodes or passes through the apertures for detection at the collector of the RFEA. ### Aperture Between Grid 2 and Grid 3 ### Theory

The RFEA is designed to have electric fields in the X direction only, it is a planar system with all grids parallel to each other. The discriminator potential at G2 is used to separate ions with different energy and acts only on the EI component of the incoming ion. The EII component of the ion energy is unaffected by the electric fields inside the sensor. At any location inside the sensor the EII component of the ion energy is identical to the EII energy component of the ion as it entered the sensor through the sampling aperture.

By varying the potential difference between G2 and G3, the acceptance angle for which ions can enter the sensor is varied. The aperture between grid 2 and grid 3 is designed to have a specific geometrical AR. Figure 3 shows a schematic representation of the aperture used to measure IEDs at the bottom of features with different ARs. The physical geometrical aperture is fixed with AR (l/w) of 0.5. The effective geometrical AR for positively charged ions can be varied by applying a potential difference (ΔV) between grid 2, which covers the entrance plane of the aperture, and grid 3, which covers the exit plane of the aperture.

Grid 2 is biased to discriminate ions based on their energy. In other words, grid 2 is used to select ions with a specific energy. As a result, the selected ions have close to zero energy when they cross grid 2 i.e. the process of selecting the ions of a particular energy involves the application of a retarding field that reduces the ion energy to almost 0 eV. In order to change the effective geometrical AR, we need to determine the ion trajectory through the aperture in terms of velocity (ν) rather than energy. The kinetic energy of the ion is equal to ½mν². Since the AR is effectively a ratio between ν⊥ and ν∥ then the ion mass (m) can be eliminated and  where the energy E has units of eV. The ion’s position (s) within the aperture (relative to the entrance plane) is given by Newton’s second law of motion, assuming the ion’s initial velocity in the perpendicular direction is zero  The time taken for an ion to reach the aperture exit plane, in terms of its perpendicular acceleration, is √(2l/a⊥) . The perpendicular velocity at the exit plane v⊥=a⊥ t=√e∆V and therefore the perpendicular acceleration is e∆V/2l. An expression for the perpendicular position can be written as The position of the ion in the parallel direction, in terms of its parallel energy, is √(E∥)t. The perpendicular position can now be written in terms of the parallel position Solving for the ion’s parallel position with respect to the exit plane gives Using basic geometric considerations, the AR is v⊥/v∥ or √(E⊥/E∥). In the same reference frame, any ion with s∥>1 will hit the side wall of the aperture. Solving for s∥=1 and substituting E⊥/AR² for E∥ gives the equivalent geometrical AR for the ion energy selected by grid 2 and the potential difference applied between grid 2 and grid 3 The software user interface (UI) allows the user to select an AR (or series of ARs) and the algorithm automatically selects ions with specified energy and applies the appropriate ΔV to keep AR constant for the duration of a specified AR scan. This graph shows the ion flux and IEDs measured for different ARs, using the technique outlined above. These measurements were taken at the powered electrode in a capacitively coupled parallel plate reactor. The driving frequency was 13.56 MHz, the working gas was oxygen and the pressure was 0.5 Pa. The drop in ion flux and the change in shape of the IED as the AR is increased is seen as expected. This implementation of the variable AR method is more convenient than the method described by Cunge et al.