Specialized Boundary Accumulators
When particles interact with a surface, it is possible for them to contribute to one or more dependent variables defined on the surface. Such variables, called accumulated variables, are described in detail in Accumulator Theory: Boundaries in Theory for the Mathematical Particle Tracing Interface.
The following specialized accumulators are available for the Charged Particle Tracing interface. In general, these accumulators are of the density type. When a particle hits a boundary mesh element, the value of the accumulated variable in that element rpb is incremented as follows:
where rpbnew is the post-collision value of the accumulated variable, rpb is the pre-collision value, R is a source term, and V is the mesh element length (in 2D) or surface area (in 3D). Each of the accumulated variables in this section is defined by the physical quantity that it represents and the expression for its source term.
Surface Charge Density Theory
The Surface Charge Density feature adds an accumulated variable for the charge density σ (SI unit: C/m2) on a set of boundaries that the particles hit. The source term for this accumulator type is
where e = 1.602176634 × 10-19 C is the elementary charge and Z (dimensionless) is the charge number of the incident particle.
Current Density Theory
The Current Density feature adds an accumulated variable for the current density of the incident particles on a set of boundaries.
If the Type is set to Normal current density, the accumulated variable is the normal current density Jn (SI unit: A/m2), and the source term is
where
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e = 1.602176634 × 10-19 C is the elementary charge,
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Z (dimensionless) is the charge number of the incident particle,
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φ (SI unit: rad) is the acute angle of incidence measured from the surface normal, and
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frel (SI unit: 1/s) is the effective frequency of release; that is, the number of real charged particles per unit time that are represented by each model particle.
A particle’s effective frequency of release is determined by the current of the particle’s release feature and the total number of particles released by the feature,
If the Type is set to Current density, one accumulated variable is defined for each component of the current density vector Js. The source term for the ith component is
where v (SI unit: m/s) is the incident particle velocity.
Heat Source Theory
The Heat Source feature adds an accumulated variable for a heat source Q (SI unit: W/m2) equal to the total kinetic energy of the incident particles. The source term is
where frel (SI unit: 1/s) is the effective frequency of release of the incident particle and E (SI unit: J) is the kinetic energy of the incident particle.
Etch Theory
The Etch feature uses accumulated variables to calculate the etch rate due to physical sputtering as energetic ions hit a surface. The dependence of the sputtering yield on the energy of particles and their angle of incidence has been described by Yin and Sawin (Ref. 11) and by Guo and Sawin (Ref. 12).
If the Plasma type is Collisionless, then the number of real ions represented by each model particle can be deduced directly from the particle release feature via the effective frequency of release. A single accumulated variable EE is defined for the etch rate on the surface (SI unit: m/s). The corresponding source term is
where
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Ms (SI unit: kg/mol) is the molar mass of the surface material,
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ρ (SI unit: kg/m3) is the density of the surface material,
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NA = 6.02214076 × 1023 1/mol is Avogadro’s constant,
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frel (SI unit: 1/s) is the effective frequency of release,
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Y() (dimensionless) is a function that indicates the dependence of the etch rate on the angle of incidence ,
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E (SI unit: J) is the particle kinetic energy,
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Eth (SI unit: J) is the threshold energy, and
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Ys (SI unit: J) is the slope of the etch yield curve.
If the Plasma type is Collisional, the effective frequency of release is no longer a reliable indicator of the incident charged particle current. Instead, incident particles in each boundary element are assigned a weighting factor Γi (SI unit: 1/(m2·s)), defined as
where the numerator n · Ji (SI unit: A/m2) is the normal current of charged particles at the surface. Usually the incident current from a collisional plasma is computed using the Plasma Module. The sum in the denominator is taken over all particles that hit the ith boundary mesh element. The etch rate in this element is then defined as
A typical angular dependence function is given by Guo and Sawin (Ref. 12) as
where the angle of incidence is measured from the surface normal. This angular dependence function is appropriate for physical sputtering of polysilicon by argon ions.