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 dp=2e^2/bv, 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 T = (dp)^2/2m = e^4/Eb^2, where
m is the electron mass and
E is the incident electron energy, given by E=(1/2) mv^2. 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 E, 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.
Resolution capability With today's
electron optics, electron-beam widths can routinely go down to a few nanometers. This is limited mainly by
aberrations and
space charge. However, the feature resolution limit is determined not by the beam size but by
forward scattering (or effective beam broadening) in the
resist, while the pitch resolution limit is determined by
secondary electron travel in the
resist. This point was driven home by a 2007 demonstration of double patterning using electron-beam lithography in the fabrication of 15 nm half-pitch zone plates. Although a 15 nm feature was resolved, a 30 nm pitch was still difficult to do due to secondary electrons scattering from the adjacent feature. The use of double patterning allowed the spacing between features to be wide enough for the secondary
electron scattering to be significantly reduced. The forward scattering can be decreased by using higher energy electrons or thinner resist, but the generation of secondary electrons is inevitable. It is now recognized that for insulating materials like
PMMA, low energy electrons can travel quite a far distance (several nm is possible). This is due to the fact that below the
ionization potential the only energy loss mechanism is mainly through
phonons and
polarons. Although the latter is basically an ionic lattice effect, polaron hopping can extend as far as 20 nm. The travel distance of secondary electrons is not a fundamentally derived physical value, but a statistical parameter often determined from many experiments or
Monte Carlo simulations down to < 1 eV. This is necessary since the energy distribution of secondary electrons peaks is well below 10 eV. Hence, the resolution limit is not usually cited as a well-fixed number as with an optical
diffraction-limited system. More recent studies have indicated that 20 nm resist thickness could be penetrated by low energy electrons (of sufficient dose) and sub-20 nm half-pitch electron-beam lithography already required double patterning. , a state-of-the-art electron multi-beam writer achieves about a 20 nm resolution.
Scattering In addition to producing secondary electrons, primary electrons from the incident beam with sufficient energy to penetrate the resist can be multiply scattered over large distances from underlying films and/or the substrate. This leads to exposure of areas at a significant distance from the desired exposure location. For thicker resists, as the primary electrons move forward, they have an increasing opportunity to scatter laterally from the beam-defined location. This scattering is called "
forward scattering". Sometimes the primary electrons are scattered at angles exceeding 90 degrees, i.e., they no longer advance further into the resist. These electrons are called "
backscattered electrons" and have the same effect as long-range
flare in optical projection systems. A large enough dose of backscattered electrons can lead to complete exposure of resist over an area much larger than defined by the beam spot.
Proximity effect The smallest features produced by electron-beam lithography have generally been isolated features, as nested features exacerbate the
proximity effect, whereby electrons from exposure of an adjacent region spill over into the exposure of the currently written feature, effectively enlarging its image, and reducing its contrast, i.e., difference between maximum and minimum intensity. Hence, nested feature resolution is harder to control. For most resists, it is difficult to go below 25 nm lines and spaces, and a limit of 20 nm lines and spaces has been found. In actuality, though, the range of secondary electron scattering is quite far, sometimes exceeding 100 nm, but becoming very significant below 30 nm. The proximity effect is also manifest by secondary electrons leaving the top surface of the resist and then returning some tens of nanometers distance away. Proximity effects (due to electron scattering) can be addressed by solving the
inverse problem and calculating the exposure function
E(x,y) that leads to a dose distribution as close as possible to the desired dose
D(x,y) when
convolved by the scattering distribution
point spread function PSF(x,y). However, it must be remembered that an error in the applied dose (e.g., from shot noise) would cause the proximity effect correction to fail. ==Charging==