Electron-Beam Lithography: The Backwards-In-Time Heat Equation in the Production of Microprocessors
Marios Andreou

• Abstract

Computer aided design is used in many material sciences, in the hardware and software sense. It is also used for fabricating integrated circuits ("electronic chips"), such as solid-state memories and microprocessors, where geometric shapes are engraved onto a silicon slab or plate (sometimes called a silicon wafer ), that will be later purposed as a processing unit for a CPU, GPU, RAM etc. These geometric shapes are created via a focused beam of electrons that draws custom shapes on a surface covered with an electron-sensitive film called a resist (exposing), with the aid of a computer, and will later form the main circuitry and logic of the board. This is called Electron-Beam Lithography or E-Beam Lithography in the business, since beams of electrons are used in the process (in general Microlithography is the term for processes that generate patterned thin films on silicon wafers). E-Beam Lithography works in the following way: First the silicon slab or substrate is covered with a electron-sensitive polymer material called a resist, and then a packet of electrons is carefully guided via a computer such that the emitted electrons cut out the desired geometric shape on the wafer, as the exposed layer will fade away. Unfortunately, electrostatic forces or external magnetic fields might be present, which would lead to electrons to be deflected due to the Lorentz force, both when penetrating the resist and reaching the substrate. This is known as electron scattering. Scattering is divided to forward and backward scattering. When the electrons interact with the resist (light-sensitive polymer) and substrate (silicon wafer) they can be scattered; forward scattering appears in both cases, while backward scattering (also known as reflection) is observed only on the latter case. In this work we look into the connection between E-Beam Lithography and the backwards in time heat equation, which is widely known to enjoy uniqueness but is unstable in any Lp norm, or that it is ill-posed for any initial condition. This connection is utilised to tackle the shape-engraving problem in the process (for two-dimensions) and at the end we provide a way of attaining a solution to this problem by virtue of Fejér’s theorem.