Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

Plasma dynamics in an inductively coupled magnetic neutral loop discharge

Magnetic neutral loop discharges (NLDs) can be operated at significantly lower pressures than conventional radio-frequency (rf) inductively coupled plasmas (ICPs). These low pressure conditions are favourable for technological applications, in particular anisotropic etching. An ICP–NLD has been designed providing excellent diagnostics access for detailed investigations of fundamental mechanisms. Spatially resolved Langmuir probe measurements have been performed in the plasma production region (NL region) as well as in the remote application region downstream from the NL region. Depending on the NL gradient two different operation modes have been observed exhibiting different opportunities for control of plasma uniformity. The efficient operation at comparatively low pressures results in ionization degrees exceeding 1%. In this regime neutral dynamics has to be considered and can influence neutral gas and process uniformity. Neutral gas depletion through elevated gas temperatures and high ionization rates have been quantified. At pressures above 0.1 Pa, gas heating is the dominant depletion mechanism. At lower pressures neutral gas is predominantly depleted through high ionization rates and rapid transport of ions by ambipolar diffusion along the magnetic field lines. Non-uniform profiles of the ionization rate can, therefore, result in localized neutral gas depletion and non-uniform processing. We have also investigated the electron dynamics within the radio-frequency cycle using phase resolved optical emission spectroscopy and Thomson scattering. In these measurements electron drift phenomena along the NL torus have been identified.

Spatial structures of plasma parameters in a magnetic neutral loop discharge

Spatial structures of plasma parameters in a radio-frequency inductively coupled magnetic neutral loop discharge are investigated under various parameter variations using spatially resolved Langmuir probe measurements. A strong coupling between the plasma production region, in the neutral loop (NL) plane, and the axially remote substrate region is observed. The two regions are connected through the separatrices and therefore, spatial profiles in the substrate region are strongly influenced by the plasma production region and the structure of the separatrices. The electron temperature in the plasma production region peaks in the centre of the NL while the maximum in electron density is shifted radially inwards due to diffusion. Details of the structures in both regions, the production region and the substrate region, are determined through the position of the NL and the gradient of the inhomogeneous magnetic field around the NL confinement region. Parameter combinations are found providing higher plasma densities and better uniformity than in common inductively coupled plasmas without applying an additional magnetic field. The uniformity can be further improved using temporal variations of the magnetic field structure.

A planar inductively coupled radio-frequency magnetic neutral loop discharge

A planar inductively coupled radio-frequency (rf) magnetic neutral loop discharge has been designed. It provides diagnostic access to both the main plasma production region as well as a remote plane for applications. Three coaxial coils are arranged to generate a specially designed inhomogeneous magnetic field structure with vanishing field along a ring in the discharge-the so-called neutral loop (NL). The plasma is generated by applying an oscillating rf electric field along the NL, induced through a four-turn, planar antenna operated at 13.56 MHz. Electron density and temperature measurements are performed under various parameter variations. Collisionless electron heating in the NL region allows plasma operation at comparatively low pressures, down to 10(-2) Pa, with a degree of ionization in the order of several per cent. Conventional plasma operation in inductive mode without applying the magnetic field is less efficient, in particular in the low pressure regime where the plasma cannot be sustained without magnetic fields.