SPIE Handbook of Microlithography, Micromachining and Microfabrication
Volume 1: Microlithography
Section 2.3 Electron-Solid Interactions
Although electron beam lithography tools are capable of forming extremely fine probes, things become more complex when the electrons hit the workpiece. As the electrons penetrate the resist, they experience many small angle scattering events (forward scattering), which tend to broaden the initial beam diameter. As the electrons penetrate through the resist into the substrate, they occasionally undergo large angle scattering events (backscattering). The backscattered electrons cause the proximity effect,  where the dose that a pattern feature receives is affected by electrons scattering from other features nearby. During this process the electrons are continuously slowing down, producing a cascade of low voltage electrons called secondary electrons.
Figure 2.10 shows some computer simulations of electron scattering in typical samples.  The combination of forward and backscattered electrons results in an energy deposition profile in the resist that is typically modeled as a sum of two Gaussian distributions, where a is the width of the forward scattering distribution, b is the width of the backscattering distribution, and ee is the intensity of the backscattered energy relative to the forward scattered energy. Fig. 2.11 shows an example of a simulated energy profile.
As the electrons penetrate the resist, some fraction of them will undergo small angle scattering events, which can result in a significantly broader beam profile at the bottom of the resist than at the fxtop. The increase in effective beam diameter in nanometers due to forward scattering is given empirically by the formula df = 0.9 (Rt / Vb)1.5, where Rt is the resist thickness in nanometers and Vb is the beam voltage in kilovolts. Forward scattering is minimized by using the thinnest possible resist and the highest available accelerating voltage.
Although it is generally best to avoid forward scattering effects when possible, in some instances they may be used to advantage. For example, it may be possible to tailor the resist sidewall angle in thick resist by adjusting the development time.  As the time increases, the resist sidewall profile will go from a positive slope, to vertical, and eventually to a negative, or retrograde, profile, which is especially desirable for pattern transfer by liftoff.
As the electrons continue to penetrate through the resist into the substrate, many of them will experience large angle scattering events. These electrons may return back through the resist at a significant distance from the incident beam, causing additional resist exposure. This is called the electron beam proximity effect. The range of the electrons (defined here as the distance a typical electron travels in the bulk material before losing all its energy) depends on both the energy of the primary electrons and the type of substrate. Fig. 2.12 shows a plot of electron range as a function of energy for three common materials. 
The fraction of electrons that are backscattered, e, is roughly independent of beam energy, although it does depend on the substrate material, with low atomic number materials giving less backscatter. Typical values of e range from 0.17 for silicon to 0.50 for tungsten and gold. Experimentally, e is only loosely related to ee, the backscatter energy deposited in the resist as modeled by a double Gaussian. Values for ee tend to be about twice e.
As the primary electrons slow down, much of their energy is dissipated in the form of secondary electrons with energies from 2 to 50 eV. They are responsible for the bulk of the actual resist exposure process. Since their range in resist is only a few nanometers, they contribute little to the proximity effect. Instead, the net result can be considered to be an effective widening of the beam diameter by roughly 10 nm. This largely accounts for the minimum practical resolution of 20 nm observed in the highest resolution electron beam systems and contributes (along with forward scattering) to the bias that is seen in positive resist systems, where the exposed features develop larger than the size they were nominally written.
A small fraction of secondary electrons may have significant energies, on the order of 1 keV. These so-called fast secondaries can contribute to the proximity effect in the range of a few tenths of a micron. Experimentally and theoretically, the distribution of these electrons can be fit well by a third Gaussian with a range intermediate between the forward scattering distribution and the backscattering distribution.
Electron scattering in resists and substrates can be modeled with reasonable accuracy by assuming that the electrons continuously slow, down as described by the Bethe equation,  while undergoing elastic scattering, as described by the screened Rutherford formula.  Since the different materials and geometries make analytic solutions difficult, Monte Carlo techniques, where a large number of random electrons are simulated, are commonly used. The input to the program contains such parameters as the electron energy, beam diameter, and film thicknesses and densities, while the output is a plot of energy deposited in the resist as a function of the distance from the center of the beam.
Curve fitting with Gaussians and other functions to the simulated energy distribution may also be employed. In order to get good statistics, the energy deposition for a large number (10,000 to 100,000) of electrons must be simulated, which can take a few minutes to an hour or so on a personal computer. Software for Monte Carlo simulation of electron irradiation is available from several sources. [24-27] Such simulations are often used to generate input parameters for proximity effect correction programs (see next section). Alternatively, experimental data can be obtained by measuring the diameter of exposed resist from a point exposure of the beam at various doses  or by measuring the linewidths of various types of test patterns such as the "tower" pattern. 
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This material is based upon work supported by the National Science Foundation under Grant No. ECCS-1542081. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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