Electron Beam Lithography - Electron Energy Deposition in Matter

Electron Energy Deposition in Matter

The primary electrons in the incident beam lose energy upon entering a material through inelastic scattering or collisions with other electrons. In such a collision the momentum transfer from the incident electron to an atomic electron can be expressed as, where b is the distance of closest approach between the electrons, and v is the incident electron velocity. The energy transferred by the collision is given by, where m is the electron mass and E is the incident electron energy, given by . By integrating over all values of T between the lowest binding energy, E₀ and the incident energy, one obtains the result that the total cross section for collision is inversely proportional to the incident energy, and proportional to 1/E₀ – 1/E. Generally, E >> E₀, so the result is essentially inversely proportional to the binding energy.

By using the same integration approach, but over the range 2E₀ to E, one obtains by comparing cross-sections that half of the inelastic collisions of the incident electrons produce electrons with kinetic energy greater than E₀. These secondary electrons are capable of breaking bonds (with binding energy E₀) at some distance away from the original collision. Additionally, they can generate additional, lower energy electrons, resulting in an electron cascade. Hence, it is important to recognize the significant contribution of secondary electrons to the spread of the energy deposition.

In general, for a molecule AB:

e− + AB → AB− → A + B−

This reaction, also known as "electron attachment" or "dissociative electron attachment" is most likely to occur after the electron has essentially slowed to a halt, since it is easiest to capture at that point. The cross-section for electron attachment is inversely proportional to electron energy at high energies, but approaches a maximum limiting value at zero energy. On the other hand, it is already known that the mean free path at the lowest energies (few to several eV or less, where dissociative attachment is significant) is well over 10 nm, thus limiting the ability to consistently achieve resolution at this scale.