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, both in the hardware and software sense. One of its main applications is in the fabrication of integrated circuits ("electronic chips"), such as solid-state memory drives and microprocessors, where geometric shapes are engraved onto a silicon slab or plate (sometimes called a silicon wafer) that is later purposed as a processing unit for a CPU, GPU, or RAM module. These geometric shapes are created with the aid of a computer by shooting a focused beam of electrons which draws custom shapes on a surface covered with an electron-sensitive film called a resist (exposing), thus forming the main circuitry and logic of the board. This process is called Electron-Beam Lithography, or E-Beam Lithography in the business, since beams of electrons are used in the aforementioned procedure. (Microlithography is the general term used for processes that generate patterned thin films on silicon wafers.) 

In detail, E-Beam Lithography works in the following way: First, the silicon slab or substrate is covered with an electron-sensitive polymer material called a resist. Then, a packet of electrons is carefully guided using a computer such that the emitted electrons cut out the desired geometric shape on the wafer, with the exposed or cut-out layer fading away. Unfortunately, electrostatic forces or external magnetic fields might be present during this process, which can potentially lead to electrons to be deflected due to Lorentz forces, both when penetrating the resist and when reaching the substrate. This is known as electron scattering and can result to a divergence from the geometric shape we want to engrave on the wafer (geometric deformities). Scattering is divided into forward and backward scattering. When the electrons interact with the resist (light-sensitive polymer) and substrate (silicon wafer) they can be scattered, with forward scattering appearing in both instances while backward scattering (also known as reflection) is observed only in the latter one. 

In these brief notes we look into the connection between E-Beam Lithography and the backwards-in-time heat equation, which is widely known to enjoy uniqueness while being unstable in any Lᵖ norm and ill-posed for any choice of initial conditions. This connection is utilised to tackle the shape-engraving problem of E-Beam Lithography in two-dimensions, where at the end we provide a way of attaining a solution to this problem by virtue of Fejér’s theorem.