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Chapter 2. Geometrical optics
2.6 Stochastic Coulomb scattering
In an earlier section, we discussed the effect of space charge on
a hypothetical test particle in the beam. The space charge was
regarded as a continuum. Approximating the current density as
uniform within the beam volume, we showed that the space charge
acts as a negative lens. In the paraxial approximation, the net result of the space charge is defocusing. The space charge lens also
has aberrations, which can be calculated in principle.
This approximation does not precisely agree with experiment,
however. The first indication of this was seen by Boersch [7], who
measured significant energy broadening in an electron beam, which
grew monotonically with current. This could not be explained
by any continuum approximation. This took on practical significance with the advent of electron beam lithography, for which
the Coulomb interaction places a limit on the useful writing current for a given resolution. This in turn limits the throughput to
values which are slow compared with optical lithography.
A beam of particles can be regarded more realistically as a collection of discrete, moving point charges, distributed randomly in
space within the beam volume. Every particle exerts a Lorentz
force (2.15) on every other particle, resulting in a random displacement of each particle. The effect becomes more pronounced
as the beam current is increased, owing to the closer proximity of
beam particles. It also becomes stronger as the interaction time is
increased, as the particles have more time to interact. The interaction time increases as the length of the beam path increases, or
the beam energy decreases.
A rough estimate of the relative strength of the interaction is obtained from the average axial spacing between particles, given by
the charge times the velocity divided by the current. In a typical
electron microscope, no more than one electron is in the column at
any given instant on average. Coulomb scattering is unimportant
in this case. In a typical electron or ion beam lithography system,
Chapter 2. Geometrical optics
2.6 Stochastic Coulomb scattering
In an earlier section, we discussed the effect of space charge on
a hypothetical test particle in the beam. The space charge was
regarded as a continuum. Approximating the current density as
uniform within the beam volume, we showed that the space charge
acts as a negative lens. In the paraxial approximation, the net result of the space charge is defocusing. The space charge lens also
has aberrations, which can be calculated in principle.
This approximation does not precisely agree with experiment,
however. The first indication of this was seen by Boersch [7], who
measured significant energy broadening in an electron beam, which
grew monotonically with current. This could not be explained
by any continuum approximation. This took on practical significance with the advent of electron beam lithography, for which
the Coulomb interaction places a limit on the useful writing current for a given resolution. This in turn limits the throughput to
values which are slow compared with optical lithography.
A beam of particles can be regarded more realistically as a collection of discrete, moving point charges, distributed randomly in
space within the beam volume. Every particle exerts a Lorentz
force (2.15) on every other particle, resulting in a random displacement of each particle. The effect becomes more pronounced
as the beam current is increased, owing to the closer proximity of
beam particles. It also becomes stronger as the interaction time is
increased, as the particles have more time to interact. The interaction time increases as the length of the beam path increases, or
the beam energy decreases.
A rough estimate of the relative strength of the interaction is obtained from the average axial spacing between particles, given by
the charge times the velocity divided by the current. In a typical
electron microscope, no more than one electron is in the column at
any given instant on average. Coulomb scattering is unimportant
in this case. In a typical electron or ion beam lithography system,