FieldEmissionBC

The electron flux boundary condition due to field emission (Based on Forbes (2006) and Forbes (2008))

Overview

FieldEmissionBC is an outflow boundary condition that assumes the electron current density is defined by field emission.

Using a Fowler-Nordheim calculation for the field emission, the electron current density is defined as

$var element = document.getElementById("moose-equation-d55eb875-ad96-447c-8359-f095751c8091");katex.render("a_{e} = \\begin{cases} 1, & \\nabla (V) \\cdot \\textbf{n} > 0\\\\ 0, & \\nabla (V) \\cdot \\textbf{n} \\leq 0\\\\ \\end{cases} \\\\[10pt] \\textbf{J}_{\\textbf{e}} \\cdot \\textbf{n} = a \\ \\phi^{-1} \\ F^{2} \\exp \\left[\\text{-}v(f) \\ b \\ \\phi^{3/2} / F \\right] \\\\[10pt] F = \\left( 1-a_{e} \\right) \\gamma \\left( \\text{-} \\nabla V \\right) \\cdot \\textbf{n} \\\\[10pt] a = 1.541434e\\text{-}6 : \\text{units of} \\ A \\ eV \\ V^{-2} \\\\[10pt] b = 6.830890e9 : \\text{units of} \\ V \\ m^{-1} \\ eV^{-3/2} \\\\[10pt] v(f) = 1 - f + \\frac{1}{6}f\\ln f \\\\[10pt] f = c \\frac{F}{\\phi^{2}} \\\\[10pt] c = 1.439964e\\text{-}9 : \\text{units of} \\ eV^{2} \\ V^{-1} \\ m", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-4800f8ab-8d00-4782-b77e-6130393d8b10");katex.render("\\textbf{J}_{\\textbf{e}}", element, {displayMode:false,throwOnError:false});$ is the electron current density, $var element = document.getElementById("moose-equation-7fa4b6f6-76a2-4767-89b3-1e1838297b48");katex.render("a", element, {displayMode:false,throwOnError:false});$ is the first Fowler–Nordheim constant, $var element = document.getElementById("moose-equation-fd46adc9-2633-4f31-afcb-87e324981944");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is the local work function, $var element = document.getElementById("moose-equation-9ec7565d-6040-4ce2-a543-9c86e22b0076");katex.render("F", element, {displayMode:false,throwOnError:false});$ is the local field, $var element = document.getElementById("moose-equation-835bd5e2-c6e6-4004-82d5-59b05fb28add");katex.render("b", element, {displayMode:false,throwOnError:false});$ is the second Fowler–Nordheim constant, $var element = document.getElementById("moose-equation-dc0f61c5-3260-42a7-a43e-ee76463a264a");katex.render("v(f)", element, {displayMode:false,throwOnError:false});$ is a correction factor that depends on the scaled barrier field ( $var element = document.getElementById("moose-equation-ee1b0856-4d25-40c7-bea1-220ea4fcdaf0");katex.render("f", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-87ec4041-78a5-49d4-a50d-dbb060e7e85b");katex.render("\\textbf{n}", element, {displayMode:false,throwOnError:false});$ is the normal vector of the boundary, $var element = document.getElementById("moose-equation-080907d7-163d-44b7-888a-7f9b15be2ff4");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ is the field enhancement factor, and $var element = document.getElementById("moose-equation-9831c9dd-237a-43e0-acb7-c34cf2f963a3");katex.render("V", element, {displayMode:false,throwOnError:false});$ is the potential. $var element = document.getElementById("moose-equation-a3372d46-2fa1-4622-8832-8d86b5999ec0");katex.render("a_{e}", element, {displayMode:false,throwOnError:false});$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. With the electron current density, the outward electron flux is defined as

$var element = document.getElementById("moose-equation-15fbcd48-233f-46e9-946c-5301b3fc2920");katex.render("\\Gamma_{i} \\cdot \\textbf{n} = \\text{sign}_{i} \\mu_{i} \\left(- \\nabla V\\right) n_{i} - D_{i} \\nabla (n_{i}) \\\\[10pt] \\Gamma_{e} \\cdot \\textbf{n} = \\frac{2 (1 - a_{e})}{1 + r} (\\text{-} (\\textbf{J}_{\\textbf{e}} \\cdot \\textbf{n})/e - \\gamma_{se} \\Gamma_{i} \\cdot \\textbf{n})", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-c894179e-badc-4bba-b1ad-4af90095c2af");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ is the outflow normal to the boundary, $var element = document.getElementById("moose-equation-c42b9cd9-cdd6-4667-9efe-20bba94f1094");katex.render("\\mu_{i}", element, {displayMode:false,throwOnError:false});$ is the mobility coefficient of the ions, $var element = document.getElementById("moose-equation-9d7c3703-a80b-4741-bb43-09137943355c");katex.render("\\text{sign}_{i}", element, {displayMode:false,throwOnError:false});$ indicates the advection behavior ( $var element = document.getElementById("moose-equation-199c2b52-7f1f-4c80-a7d4-3c9b80a0398e");katex.render("\\text{+}1", element, {displayMode:false,throwOnError:false});$ for positively charged species and $var element = document.getElementById("moose-equation-77aaa877-79e4-4b7d-9987-bb9052e87a48");katex.render("\\text{-}1", element, {displayMode:false,throwOnError:false});$ for negatively charged species), $var element = document.getElementById("moose-equation-6e8f7166-694f-4ee7-8e6d-e1f140fad584");katex.render("n_{i}", element, {displayMode:false,throwOnError:false});$ is the ion density, $var element = document.getElementById("moose-equation-22d701a6-f5ae-4e76-96ae-a97aad7862bb");katex.render("D_{i}", element, {displayMode:false,throwOnError:false});$ is the diffusion coefficient of ions, $var element = document.getElementById("moose-equation-8f438d4d-b5bd-49e4-92b6-e441df9ba77d");katex.render("e", element, {displayMode:false,throwOnError:false});$ is the elemental charge, and $var element = document.getElementById("moose-equation-264bcf95-a00b-4513-95ae-6c3d5dd8217d");katex.render("\\gamma_{se}", element, {displayMode:false,throwOnError:false});$ is the ion induced secondary electron emission coefficient. $var element = document.getElementById("moose-equation-8db649ed-c68a-4557-90e8-24273445c5fc");katex.render("r", element, {displayMode:false,throwOnError:false});$ is defined as the fraction of particles reflected by the surface.

When converting the density to log form and applying a scaling factor of the mesh and voltage, the strong form is given by

$var element = document.getElementById("moose-equation-e7d3e6c6-a628-4f93-879c-f6be4af5e53f");katex.render("F = \\left( 1-a \\right) \\gamma \\left( \\text{-} \\nabla V / l_{c} \\right) \\cdot \\textbf{n} \\\\[10pt] a = 1.541434e\\text{-}6 * V_{c}^{2} \\\\[10pt] b = 6.830890e9 / V_{c} \\\\[10pt] c = 1.439964e\\text{-}9 * V_{c}\\\\[10pt] \\Gamma_{i} \\cdot \\textbf{n} = \\text{sign}_{i} \\mu_{i} \\left( - \\nabla V / l_{c}\\right) \\exp(N_{i}) - D_{i} \\exp(N_{i}) \\nabla (N_{i} / l_{c}) \\\\[10pt] \\Gamma_{e} \\cdot \\textbf{n} = \\frac{2 (1 - a)}{1 + r} (\\text{-} (\\textbf{J}_{\\textbf{e}} \\cdot \\textbf{n})/ (e / N_{A}) - \\gamma_{se} \\Gamma_{i} \\cdot \\textbf{n})", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-21a1ed02-b5a0-436b-8522-834bad34f609");katex.render("N_{i}", element, {displayMode:false,throwOnError:false});$ is the molar density of the species in log form, $var element = document.getElementById("moose-equation-3efc2105-4af9-4027-bad4-4df029aea620");katex.render("N_{A}", element, {displayMode:false,throwOnError:false});$ is Avogadro's number, $var element = document.getElementById("moose-equation-64c0a4c2-4690-4db7-bc83-e62b11e3215e");katex.render("V_{c}", element, {displayMode:false,throwOnError:false});$ is the scaling factor of the potential, and $var element = document.getElementById("moose-equation-06ee5454-c27b-49d5-941b-781da7d54f34");katex.render("l_{c}", element, {displayMode:false,throwOnError:false});$ is the scaling factor of the mesh.

Example Input File Syntax

[BCs]
  [FieldEmission_left]
    type = FieldEmissionBC
    variable = em
    boundary = 'left'
    potential = potential
    ip = Arp
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
  []
[]

Input Parameters

boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
emission_coeffsThe species dependent secondary electron emission coefficients for this boundary
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
Controllable:No
Description:The species dependent secondary electron emission coefficients for this boundary
ionsA list of ion densities in log form
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:A list of ion densities in log form
position_unitsUnits of position.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Units of position.
potentialThe electric potential
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The electric potential
potential_unitsThe potential units.
C++ Type:std::string
Unit:(no unit assumed)
Controllable:No
Description:The potential units.
rThe reflection coefficient
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The reflection coefficient
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
relaxFalseUse relaxation for emission.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Use relaxation for emission.
tau1e-09The time constant for ramping the boundary condition.
Default:1e-09
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The time constant for ramping the boundary condition.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Unit:(no unit assumed)
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

Input Files

(test/tests/field_emission/field_emission.i)

References

Richard G Forbes. Simple good approximations for the special elliptic functions in standard fowler-nordheim tunneling theory for a schottky-nordheim barrier. Applied physics letters, 2006. doi:10.1063/1.2354582.

@article{forbes2006simple,
    author = "Forbes, Richard G",
    title = "Simple good approximations for the special elliptic functions in standard Fowler-Nordheim tunneling theory for a Schottky-Nordheim barrier",
    journal = "Applied physics letters",
    volume = "89",
    number = "11",
    year = "2006",
    publisher = "AIP Publishing",
    doi = "10.1063/1.2354582"
}

Richard G Forbes. Physics of generalized fowler-nordheim-type equations. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena, 26(2):788–793, 2008. doi:10.1116/1.2827505.

@article{forbes2008physics,
    author = "Forbes, Richard G",
    title = "Physics of generalized Fowler-Nordheim-type equations",
    journal = "Journal of Vacuum Science \\& Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena",
    volume = "26",
    number = "2",
    pages = "788--793",
    year = "2008",
    publisher = "AIP Publishing",
    doi = "10.1116/1.2827505"
}

(test/tests/field_emission/field_emission.i)

dom0Scale = 1
dom0Size = 5E-6 #m
vhigh = -0.15 #kV

[GlobalParams]
  offset = 20
  potential_units = kV
  # potential_units = V
  use_moles = true
[]

[Mesh]
  [file]
    type = FileMeshGenerator
    file = 'micro_fe.msh'
  []
  [left]
    type = SideSetsFromNormalsGenerator
    normals = '-1 0 0'
    new_boundary = 'left'
    input = file
  []
  [right]
    type = SideSetsFromNormalsGenerator
    normals = '1 0 0'
    new_boundary = 'right'
    input = left
  []
[]

[Problem]
  type = FEProblem
  # kernel_coverage_check = false
[]

[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  end_time = 6E-6
  petsc_options = '-snes_converged_reason -snes_linesearch_monitor'
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -ksp_type -snes_linesearch_minlambda'
  petsc_options_value = 'lu NONZERO 1.e-10 preonly 1e-3'
  nl_rel_tol = 1e-4
  nl_abs_tol = 7.6e-5
  dtmin = 1e-15
  dtmax = 0.01E-7
  [TimeSteppers]
    [Adaptive]
      type = IterationAdaptiveDT
      cutback_factor = 0.4
      dt = 1e-12
      growth_factor = 1.2
      optimal_iterations = 15
    []
  []
[]

[Outputs]
  perf_graph = true
  print_linear_residuals = false
  [out]
    type = Exodus
    #       execute_on = 'final'
  []
[]

[Debug]
  show_var_residual_norms = true
[]

[UserObjects]
  [data_provider]
    type = ProvideMobility
    electrode_area = 5.02e-7 # Formerly 3.14e-6
    ballast_resist = 1e6
    e = 1.6e-19
  []
[]

[Kernels]
  [em_time_deriv]
    type = ElectronTimeDerivative
    variable = em
    block = 0
  []
  [em_advection]
    type = EFieldAdvection
    variable = em
    potential = potential
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_diffusion]
    type = CoeffDiffusion
    variable = em
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_ionization]
    type = ElectronsFromIonization
    em = em
    variable = em
    potential = potential
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_log_stabilization]
    type = LogStabilizationMoles
    variable = em
    block = 0
  []
  # [em_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = em
  #               potential = potential
  #               block = 0
  # []

  [potential_diffusion_dom1]
    type = CoeffDiffusionLin
    variable = potential
    block = 0
    position_units = ${dom0Scale}
  []

  [Arp_charge_source]
    type = ChargeSourceMoles_KV
    variable = potential
    charged = Arp
    block = 0
  []
  [em_charge_source]
    type = ChargeSourceMoles_KV
    variable = potential
    charged = em
    block = 0
  []

  [Arp_time_deriv]
    type = ElectronTimeDerivative
    variable = Arp
    block = 0
  []
  [Arp_advection]
    type = EFieldAdvection
    variable = Arp
    potential = potential
    position_units = ${dom0Scale}
    block = 0
  []
  [Arp_diffusion]
    type = CoeffDiffusion
    variable = Arp
    block = 0
    position_units = ${dom0Scale}
  []
  [Arp_ionization]
    type = IonsFromIonization
    variable = Arp
    potential = potential
    em = em
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [Arp_log_stabilization]
    type = LogStabilizationMoles
    variable = Arp
    block = 0
  []
  # [Arp_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = Arp
  #               potential = potential
  #               block = 0
  # []

  [mean_en_time_deriv]
    type = ElectronTimeDerivative
    variable = mean_en
    block = 0
  []
  [mean_en_advection]
    type = EFieldAdvection
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_diffusion]
    type = CoeffDiffusion
    variable = mean_en
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_joule_heating]
    type = JouleHeating
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_ionization]
    type = ElectronEnergyLossFromIonization
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_elastic]
    type = ElectronEnergyLossFromElastic
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_excitation]
    type = ElectronEnergyLossFromExcitation
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_log_stabilization]
    type = LogStabilizationMoles
    variable = mean_en
    block = 0
    offset = 15
  []
  # [mean_en_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = mean_en
  #               potential = potential
  #               block = 0
  # []
[]

[Variables]
  [potential]
  []
  [em]
    block = 0
  []
  [Arp]
    block = 0
  []
  [mean_en]
    block = 0
  []

[]

[AuxVariables]
  [e_temp]
    block = 0
    order = CONSTANT
    family = MONOMIAL
  []
  [x]
    order = CONSTANT
    family = MONOMIAL
  []
  [x_node]
  []
  [rho]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [em_lin]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Arp_lin]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Efield]
    order = CONSTANT
    family = MONOMIAL
  []
  [Current_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Current_Arp]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [tot_gas_current]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [EFieldAdvAux_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [DiffusiveFlux_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [PowerDep_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [PowerDep_Arp]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_el]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_ex]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_iz]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
[]

[AuxKernels]
  [PowerDep_em]
    type = PowerDep
    density_log = em
    potential = potential
    art_diff = false
    potential_units = kV
    variable = PowerDep_em
    position_units = ${dom0Scale}
    block = 0
  []
  [PowerDep_Arp]
    type = PowerDep
    density_log = Arp
    potential = potential
    art_diff = false
    potential_units = kV
    variable = PowerDep_Arp
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_el]
    type = ProcRate
    em = em
    potential = potential
    proc = el
    variable = ProcRate_el
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_ex]
    type = ProcRate
    em = em
    potential = potential
    proc = ex
    variable = ProcRate_ex
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_iz]
    type = ProcRate
    em = em
    potential = potential
    proc = iz
    variable = ProcRate_iz
    position_units = ${dom0Scale}
    block = 0
  []
  [e_temp]
    type = ElectronTemperature
    variable = e_temp
    electron_density = em
    mean_en = mean_en
    block = 0
  []
  [x_g]
    type = Position
    variable = x
    position_units = ${dom0Scale}
    block = 0
  []
  [x_ng]
    type = Position
    variable = x_node
    position_units = ${dom0Scale}
    block = 0
  []
  [rho]
    type = ParsedAux
    variable = rho
    coupled_variables = 'em_lin Arp_lin'
    expression = 'Arp_lin - em_lin'
    execute_on = 'timestep_end'
    block = 0
  []
  [tot_gas_current]
    type = ParsedAux
    variable = tot_gas_current
    coupled_variables = 'Current_em Current_Arp'
    expression = 'Current_em + Current_Arp'
    execute_on = 'timestep_end'
    block = 0
  []
  [em_lin]
    type = Density
    #               convert_moles = true
    variable = em_lin
    density_log = em
    block = 0
  []
  [Arp_lin]
    type = Density
    #               convert_moles = true
    variable = Arp_lin
    density_log = Arp
    block = 0
  []
  [Efield_g]
    type = Efield
    component = 0
    potential = potential
    variable = Efield
    position_units = ${dom0Scale}
    block = 0
  []
  [Current_em]
    type = Current
    potential = potential
    density_log = em
    variable = Current_em
    art_diff = false
    block = 0
    position_units = ${dom0Scale}
  []
  [Current_Arp]
    type = Current
    potential = potential
    density_log = Arp
    variable = Current_Arp
    art_diff = false
    block = 0
    position_units = ${dom0Scale}
  []
  [EFieldAdvAux_em]
    type = EFieldAdvAux
    potential = potential
    density_log = em
    variable = EFieldAdvAux_em
    block = 0
    position_units = ${dom0Scale}
  []
  [DiffusiveFlux_em]
    type = DiffusiveFlux
    density_log = em
    variable = DiffusiveFlux_em
    block = 0
    position_units = ${dom0Scale}
  []
[]

[BCs]
  [potential_left]
    type = NeumannCircuitVoltageMoles_KV
    variable = potential
    boundary = left
    function = potential_bc_func
    ip = Arp
    data_provider = data_provider
    em = em
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
    emission_coeffs = 0.05
  []
  [potential_dirichlet_right]
    type = DirichletBC
    variable = potential
    boundary = right
    value = 0
  []
  [em_physical_right]
    type = HagelaarElectronBC
    variable = em
    boundary = right
    potential = potential
    mean_en = mean_en
    r = 0.99
    position_units = ${dom0Scale}
  []

  [Arp_physical_right_diffusion]
    type = HagelaarIonDiffusionBC
    variable = Arp
    boundary = right
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_right_advection]
    type = HagelaarIonAdvectionBC
    variable = Arp
    boundary = right
    potential = potential
    r = 0
    position_units = ${dom0Scale}
  []
  [mean_en_physical_right]
    type = HagelaarEnergyBC
    variable = mean_en
    boundary = right
    potential = potential
    em = em
    ip = Arp
    r = 0.99
    position_units = ${dom0Scale}
  []
  [em_physical_left]
    type = HagelaarElectronBC
    variable = em
    boundary = 'left'
    potential = potential
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
  []
  #       [sec_electrons_left]
  #               type = SecondaryElectronBC
  #               variable = em
  #               boundary = 'left'
  #               potential = potential
  #               ip = Arp
  #               mean_en = mean_en
  #               r = 0
  #               position_units = ${dom0Scale}
  #       []
  [FieldEmission_left]
    type = FieldEmissionBC
    variable = em
    boundary = 'left'
    potential = potential
    ip = Arp
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_left_diffusion]
    type = HagelaarIonDiffusionBC
    variable = Arp
    boundary = 'left'
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_left_advection]
    type = HagelaarIonAdvectionBC
    variable = Arp
    boundary = 'left'
    potential = potential
    r = 0
    position_units = ${dom0Scale}
  []
  [mean_en_physical_left]
    type = HagelaarEnergyBC
    variable = mean_en
    boundary = 'left'
    potential = potential
    em = em
    ip = Arp
    r = 0
    position_units = ${dom0Scale}
  []
[]

[ICs]
  [em_ic]
    type = ConstantIC
    variable = em
    value = -21
    block = 0
  []
  [Arp_ic]
    type = ConstantIC
    variable = Arp
    value = -21
    block = 0
  []
  [mean_en_ic]
    type = ConstantIC
    variable = mean_en
    value = -20
    block = 0
  []

  [potential_ic]
    type = FunctionIC
    variable = potential
    function = potential_ic_func
  []
  # [em_ic]
  #               type = RandomIC
  #               variable = em
  #               block = 0
  # []
  # [Arp_ic]
  #               type = RandomIC
  #               variable = Arp
  #               block = 0
  # []
  # [mean_en_ic]
  #               type = RandomIC
  #               variable = mean_en
  #               block = 0
  # []
  # [potential_ic]
  #               type = RandomIC
  #               variable = potential
  # []
[]

[Functions]
  [potential_bc_func]
    type = ParsedFunction
    symbol_names = 'period dutyCycle riseTime VHigh VLow'
    symbol_values = '3E-6 0.1 5E-7 ${vhigh} -0.001'
    expression = 'if((t % period) < dutyCycle*period           , VHigh                                                                  ,
                                 if((t % period) < dutyCycle*period + riseTime, ((VLow - VHigh)/riseTime) * ((t % period) - period * dutyCycle) + VHigh,
                                 if((t % period) < period - riseTime          , VLow                                                                                                                               ,
                                 if((t % period) < period                                         , ((VHigh - VLow)/riseTime) * ((t % period) - period) + VHigh            ,
                                        0))))'
  []
  [potential_ic_func]
    type = ParsedFunction
    expression = '-${vhigh} * (${dom0Size} - x) / ${dom0Size}'
  []
  [cathode_temperature]
    type = ParsedFunction
    expression = 1500
  []
[]

[Materials]
  [gas_block]
    type = Gas
    interp_trans_coeffs = true
    interp_elastic_coeff = true
    ramp_trans_coeffs = false
    em = em
    potential = potential
    ip = Arp
    mean_en = mean_en
    user_se_coeff = 0.05
    user_work_function = 4.55 # eV
    user_field_enhancement = 55
    property_tables_file = td_argon_mean_en.txt
    block = 0
  []
[]

(test/tests/field_emission/field_emission.i)

dom0Scale = 1
dom0Size = 5E-6 #m
vhigh = -0.15 #kV

[GlobalParams]
  offset = 20
  potential_units = kV
  # potential_units = V
  use_moles = true
[]

[Mesh]
  [file]
    type = FileMeshGenerator
    file = 'micro_fe.msh'
  []
  [left]
    type = SideSetsFromNormalsGenerator
    normals = '-1 0 0'
    new_boundary = 'left'
    input = file
  []
  [right]
    type = SideSetsFromNormalsGenerator
    normals = '1 0 0'
    new_boundary = 'right'
    input = left
  []
[]

[Problem]
  type = FEProblem
  # kernel_coverage_check = false
[]

[Preconditioning]
  [smp]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  end_time = 6E-6
  petsc_options = '-snes_converged_reason -snes_linesearch_monitor'
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -ksp_type -snes_linesearch_minlambda'
  petsc_options_value = 'lu NONZERO 1.e-10 preonly 1e-3'
  nl_rel_tol = 1e-4
  nl_abs_tol = 7.6e-5
  dtmin = 1e-15
  dtmax = 0.01E-7
  [TimeSteppers]
    [Adaptive]
      type = IterationAdaptiveDT
      cutback_factor = 0.4
      dt = 1e-12
      growth_factor = 1.2
      optimal_iterations = 15
    []
  []
[]

[Outputs]
  perf_graph = true
  print_linear_residuals = false
  [out]
    type = Exodus
    #       execute_on = 'final'
  []
[]

[Debug]
  show_var_residual_norms = true
[]

[UserObjects]
  [data_provider]
    type = ProvideMobility
    electrode_area = 5.02e-7 # Formerly 3.14e-6
    ballast_resist = 1e6
    e = 1.6e-19
  []
[]

[Kernels]
  [em_time_deriv]
    type = ElectronTimeDerivative
    variable = em
    block = 0
  []
  [em_advection]
    type = EFieldAdvection
    variable = em
    potential = potential
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_diffusion]
    type = CoeffDiffusion
    variable = em
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_ionization]
    type = ElectronsFromIonization
    em = em
    variable = em
    potential = potential
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [em_log_stabilization]
    type = LogStabilizationMoles
    variable = em
    block = 0
  []
  # [em_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = em
  #               potential = potential
  #               block = 0
  # []

  [potential_diffusion_dom1]
    type = CoeffDiffusionLin
    variable = potential
    block = 0
    position_units = ${dom0Scale}
  []

  [Arp_charge_source]
    type = ChargeSourceMoles_KV
    variable = potential
    charged = Arp
    block = 0
  []
  [em_charge_source]
    type = ChargeSourceMoles_KV
    variable = potential
    charged = em
    block = 0
  []

  [Arp_time_deriv]
    type = ElectronTimeDerivative
    variable = Arp
    block = 0
  []
  [Arp_advection]
    type = EFieldAdvection
    variable = Arp
    potential = potential
    position_units = ${dom0Scale}
    block = 0
  []
  [Arp_diffusion]
    type = CoeffDiffusion
    variable = Arp
    block = 0
    position_units = ${dom0Scale}
  []
  [Arp_ionization]
    type = IonsFromIonization
    variable = Arp
    potential = potential
    em = em
    mean_en = mean_en
    block = 0
    position_units = ${dom0Scale}
  []
  [Arp_log_stabilization]
    type = LogStabilizationMoles
    variable = Arp
    block = 0
  []
  # [Arp_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = Arp
  #               potential = potential
  #               block = 0
  # []

  [mean_en_time_deriv]
    type = ElectronTimeDerivative
    variable = mean_en
    block = 0
  []
  [mean_en_advection]
    type = EFieldAdvection
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_diffusion]
    type = CoeffDiffusion
    variable = mean_en
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_joule_heating]
    type = JouleHeating
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_ionization]
    type = ElectronEnergyLossFromIonization
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_elastic]
    type = ElectronEnergyLossFromElastic
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_excitation]
    type = ElectronEnergyLossFromExcitation
    variable = mean_en
    potential = potential
    em = em
    block = 0
    position_units = ${dom0Scale}
  []
  [mean_en_log_stabilization]
    type = LogStabilizationMoles
    variable = mean_en
    block = 0
    offset = 15
  []
  # [mean_en_advection_stabilization]
  #               type = EFieldArtDiff
  #               variable = mean_en
  #               potential = potential
  #               block = 0
  # []
[]

[Variables]
  [potential]
  []
  [em]
    block = 0
  []
  [Arp]
    block = 0
  []
  [mean_en]
    block = 0
  []

[]

[AuxVariables]
  [e_temp]
    block = 0
    order = CONSTANT
    family = MONOMIAL
  []
  [x]
    order = CONSTANT
    family = MONOMIAL
  []
  [x_node]
  []
  [rho]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [em_lin]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Arp_lin]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Efield]
    order = CONSTANT
    family = MONOMIAL
  []
  [Current_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [Current_Arp]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [tot_gas_current]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [EFieldAdvAux_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [DiffusiveFlux_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [PowerDep_em]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [PowerDep_Arp]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_el]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_ex]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
  [ProcRate_iz]
    order = CONSTANT
    family = MONOMIAL
    block = 0
  []
[]

[AuxKernels]
  [PowerDep_em]
    type = PowerDep
    density_log = em
    potential = potential
    art_diff = false
    potential_units = kV
    variable = PowerDep_em
    position_units = ${dom0Scale}
    block = 0
  []
  [PowerDep_Arp]
    type = PowerDep
    density_log = Arp
    potential = potential
    art_diff = false
    potential_units = kV
    variable = PowerDep_Arp
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_el]
    type = ProcRate
    em = em
    potential = potential
    proc = el
    variable = ProcRate_el
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_ex]
    type = ProcRate
    em = em
    potential = potential
    proc = ex
    variable = ProcRate_ex
    position_units = ${dom0Scale}
    block = 0
  []
  [ProcRate_iz]
    type = ProcRate
    em = em
    potential = potential
    proc = iz
    variable = ProcRate_iz
    position_units = ${dom0Scale}
    block = 0
  []
  [e_temp]
    type = ElectronTemperature
    variable = e_temp
    electron_density = em
    mean_en = mean_en
    block = 0
  []
  [x_g]
    type = Position
    variable = x
    position_units = ${dom0Scale}
    block = 0
  []
  [x_ng]
    type = Position
    variable = x_node
    position_units = ${dom0Scale}
    block = 0
  []
  [rho]
    type = ParsedAux
    variable = rho
    coupled_variables = 'em_lin Arp_lin'
    expression = 'Arp_lin - em_lin'
    execute_on = 'timestep_end'
    block = 0
  []
  [tot_gas_current]
    type = ParsedAux
    variable = tot_gas_current
    coupled_variables = 'Current_em Current_Arp'
    expression = 'Current_em + Current_Arp'
    execute_on = 'timestep_end'
    block = 0
  []
  [em_lin]
    type = Density
    #               convert_moles = true
    variable = em_lin
    density_log = em
    block = 0
  []
  [Arp_lin]
    type = Density
    #               convert_moles = true
    variable = Arp_lin
    density_log = Arp
    block = 0
  []
  [Efield_g]
    type = Efield
    component = 0
    potential = potential
    variable = Efield
    position_units = ${dom0Scale}
    block = 0
  []
  [Current_em]
    type = Current
    potential = potential
    density_log = em
    variable = Current_em
    art_diff = false
    block = 0
    position_units = ${dom0Scale}
  []
  [Current_Arp]
    type = Current
    potential = potential
    density_log = Arp
    variable = Current_Arp
    art_diff = false
    block = 0
    position_units = ${dom0Scale}
  []
  [EFieldAdvAux_em]
    type = EFieldAdvAux
    potential = potential
    density_log = em
    variable = EFieldAdvAux_em
    block = 0
    position_units = ${dom0Scale}
  []
  [DiffusiveFlux_em]
    type = DiffusiveFlux
    density_log = em
    variable = DiffusiveFlux_em
    block = 0
    position_units = ${dom0Scale}
  []
[]

[BCs]
  [potential_left]
    type = NeumannCircuitVoltageMoles_KV
    variable = potential
    boundary = left
    function = potential_bc_func
    ip = Arp
    data_provider = data_provider
    em = em
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
    emission_coeffs = 0.05
  []
  [potential_dirichlet_right]
    type = DirichletBC
    variable = potential
    boundary = right
    value = 0
  []
  [em_physical_right]
    type = HagelaarElectronBC
    variable = em
    boundary = right
    potential = potential
    mean_en = mean_en
    r = 0.99
    position_units = ${dom0Scale}
  []

  [Arp_physical_right_diffusion]
    type = HagelaarIonDiffusionBC
    variable = Arp
    boundary = right
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_right_advection]
    type = HagelaarIonAdvectionBC
    variable = Arp
    boundary = right
    potential = potential
    r = 0
    position_units = ${dom0Scale}
  []
  [mean_en_physical_right]
    type = HagelaarEnergyBC
    variable = mean_en
    boundary = right
    potential = potential
    em = em
    ip = Arp
    r = 0.99
    position_units = ${dom0Scale}
  []
  [em_physical_left]
    type = HagelaarElectronBC
    variable = em
    boundary = 'left'
    potential = potential
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
  []
  #       [sec_electrons_left]
  #               type = SecondaryElectronBC
  #               variable = em
  #               boundary = 'left'
  #               potential = potential
  #               ip = Arp
  #               mean_en = mean_en
  #               r = 0
  #               position_units = ${dom0Scale}
  #       []
  [FieldEmission_left]
    type = FieldEmissionBC
    variable = em
    boundary = 'left'
    potential = potential
    ip = Arp
    mean_en = mean_en
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_left_diffusion]
    type = HagelaarIonDiffusionBC
    variable = Arp
    boundary = 'left'
    r = 0
    position_units = ${dom0Scale}
  []
  [Arp_physical_left_advection]
    type = HagelaarIonAdvectionBC
    variable = Arp
    boundary = 'left'
    potential = potential
    r = 0
    position_units = ${dom0Scale}
  []
  [mean_en_physical_left]
    type = HagelaarEnergyBC
    variable = mean_en
    boundary = 'left'
    potential = potential
    em = em
    ip = Arp
    r = 0
    position_units = ${dom0Scale}
  []
[]

[ICs]
  [em_ic]
    type = ConstantIC
    variable = em
    value = -21
    block = 0
  []
  [Arp_ic]
    type = ConstantIC
    variable = Arp
    value = -21
    block = 0
  []
  [mean_en_ic]
    type = ConstantIC
    variable = mean_en
    value = -20
    block = 0
  []

  [potential_ic]
    type = FunctionIC
    variable = potential
    function = potential_ic_func
  []
  # [em_ic]
  #               type = RandomIC
  #               variable = em
  #               block = 0
  # []
  # [Arp_ic]
  #               type = RandomIC
  #               variable = Arp
  #               block = 0
  # []
  # [mean_en_ic]
  #               type = RandomIC
  #               variable = mean_en
  #               block = 0
  # []
  # [potential_ic]
  #               type = RandomIC
  #               variable = potential
  # []
[]

[Functions]
  [potential_bc_func]
    type = ParsedFunction
    symbol_names = 'period dutyCycle riseTime VHigh VLow'
    symbol_values = '3E-6 0.1 5E-7 ${vhigh} -0.001'
    expression = 'if((t % period) < dutyCycle*period           , VHigh                                                                  ,
                                 if((t % period) < dutyCycle*period + riseTime, ((VLow - VHigh)/riseTime) * ((t % period) - period * dutyCycle) + VHigh,
                                 if((t % period) < period - riseTime          , VLow                                                                                                                               ,
                                 if((t % period) < period                                         , ((VHigh - VLow)/riseTime) * ((t % period) - period) + VHigh            ,
                                        0))))'
  []
  [potential_ic_func]
    type = ParsedFunction
    expression = '-${vhigh} * (${dom0Size} - x) / ${dom0Size}'
  []
  [cathode_temperature]
    type = ParsedFunction
    expression = 1500
  []
[]

[Materials]
  [gas_block]
    type = Gas
    interp_trans_coeffs = true
    interp_elastic_coeff = true
    ramp_trans_coeffs = false
    em = em
    potential = potential
    ip = Arp
    mean_en = mean_en
    user_se_coeff = 0.05
    user_work_function = 4.55 # eV
    user_field_enhancement = 55
    property_tables_file = td_argon_mean_en.txt
    block = 0
  []
[]

Overview
Example Input File Syntax
Input Parameters
Input Files
References