fieldEmitterVelGen

fieldEmitterVelGen

Velocity generator that fixes the components of the velocity-based values of a given set of space-time functions. The following references are relevant to electron field emitters:

“Electron emission theory and its application: Fowler–Nordheim equation and beyond,” J. Vac. Sci. Technol. B, 21:1528-44, 2003.
“A tutorial on electron sources.” IEEE Transactions on Plasma Science 46.6 (2018): 1881-1899.
“Electron emission contributions to dark current and its relation to microscopic field enhancement and heating in accelerator structures.” Physical Review Special Topics-Accelerators and Beams 11.8 (2008)

fieldEmitterVelGen Parameters

emitVelocityIsLocal (boolean): Specifies the orientation of the component references in the block. When false, components refer to the overall coordinate system axes, and when true, components refer to local orientation of the emission surface. A setting of true is commonly used when emitting from a cut-cell surface, in order to insure normal emission. In this case, the normal direction is inward, towards the surface, so that the sign of the component0 velocity is typically negative to insure outward flow of particles. By default, emission velocity is zero, while the emission velocity can be specified with <STFunc component0/1/2>

work\_function (real): Work function for the metal.

alpha (real): Normalized offset distance where alpha = 1 corresponds to the length of the hypotenuse of one cell. If alpha = 1 the electric field is sampled normal to the boundary a distance alpha*H from the emission point (where H is the hypotenuse). This factor needed since the field at the boundary is smaller than the field just inside the boundary. The field just inside the boundary is the correct field.

temperature (real): Temperature of the metal.

multiplier (real): Multiplication factor to enhance the current.

emitterType (string)

One of:

Richardson-Dushman
Fowler-Nordheim
txemit

Richardson-Dushman Emitter

Emitter that evaluates the equation:

(21)\[J = M \frac{4 \pi m e}{h^3} T^2 \exp\left({-\frac{W}{T}}\right),\]

where:

\(J\) is the resulting magnitude of the current density, \(M\) is the multiplier, \(m\) is the electron mass, \(e\) is the electron charge, \(h\) is Planck’s constant, \(T\) is the temperature in appropriate units, and \(W\) is given by

(22)\[W = W_0 - \frac{e}{4 \pi \epsilon_0} | {\bf E} f|,\]

where:

\({\bf E}\) is the surface electric field, \(f\) is the enhancement factor, and \(W_0\) is the work function.

Richardson-Dushman Emitter Parameters:

field\_enhancement (double, default = 1): Multiplies the measured electric field.

multiplier (double, default = 1): Multiplies the resulting output current.

work\_function (double, default = 4.5): Minimum energy necessary to extract an electron from the surface; measured in electron volts.

temperature (double): Temperature of the surface in Kelvin.

In the Richardson-Dushman emitter, the work function may be specified using an STFunc block named workFunction. The work function defined in the STFunc block overwrites the value of the work_function keyword.

Syntax to Specify the Work Function as an STFunc Block:

<VelocityGenerator emitVelGen>
  kind = fieldEmitterVelGen
  emitVelocityIsLocal = false
  work_function = 4.5
  alpha = 0.1
  temperature = 2000.0
  field_enhancement = 1.0
  multiplier = 1.0
  emitterType = Richardson-Dushman
</VelocityGenerator>

Fowler-Nordheim Emitter

Fowler-Nordheim emitters are a class of field emitters. For algorithms that compute emission due to thermal-field and in the photo regime, implementations are based on papers by K.L. Jensen: 1. JAP 102, 024911 (2007), and 2. PR-STAB 11, 081001 (2008) with correction made to eq. 40 (from 1) for \(C_{FN}= 2*fi*\sqrt{2*mo*Phi}*t(y)/(hbar*Field)\).

Field emission current density, \(J_{FN}\), is determined by:

(23)\[J_{FN} = \frac{A_{FN} \left( \beta_{FN} E_p \right)^2 }{\phi_w} \exp\left({-\frac{B_{FN} v(y) \phi_w^{3/2}}{\beta_{FN} E_p}}\right),\]

where:

\(E_{p}\) is the perpendicular component of the electric field in V/m, \(\beta_{FN}\) is the field enhancement factor, \(\phi_{w}\) is the work function of the field-emitting material surface, \(v(y)\) has the form

(24)\[v(y) = Pfn - C\_v\_FN y^2 + (1/3)y^2 \ln(y)\]

where:

\(y\) is given by:

(25)\[y = \frac{C_y E_y^{1/2}}{\phi_w},\]

and:

\(A_{FN}\) , \(B_{FN}\) , \(C_{v}\) , and \(C_{y}\) are coefficients of the Fowler-Nordheim field emission model.

and:

\(Pfn\) is actually an input parameter that defaults to 0.95 (it’s described as the “Nordheim coefficient”), and \(C\_v\_FN\) is a parameter described below.

Fowler-Nordheim Parameters

beta\_FN (real, default = 1.0): Field enhancement factor \(\beta_{FN}\) of the Fowler-Nordheim field emission model.

A\_FN (real, default = 1.5414e-6): Coefficient \(A_{FN}\) of the Fowler-Nordheim field emission model.

B\_FN (real, default = 6.8308e9): Coefficient \(B_{FN}\) of the Fowler-Nordheim field emission model.

C\_v\_FN (real, default = 0.0): Coefficient \(C_{v}\) of the Fowler-Nordheim field emission model.

C\_y\_FN (real, default value = 3.79e-5): Coefficient \(C_{y}\) of the Fowler-Nordheim field emission model.

Photo-emission Emitter

Field emitter using the txphysics txemit field-thermal and photo emission emitters.

txemit Photo-emission Parameters:

mu (real, default = 1.8): Fermi Level of material.

laser (boolean, default = false): True or false. Whether to use a laser for photo emission.

beta\_FN (real, default = 1.0): Field enhancement factor.

txemit Photo-emission Required Extra Parameters for laser=true:

laserPulseStartTime (float, default = 0.0): Starting time for calculating intensity of pulse at the cathode. Required for use with photo-emission Laser pulse.

laserPulseStopTime (float): Ending time for calculating intensity of pulse at the cathode. Required for use with photo-emission Laser pulse.

laserPulseOrigin (float vector): Origin from which the pulse is launched. The origin is defined at the pulse maximum value at the startTime. Required for use with photo-emission Laser pulse.

laserPulseIntensityEnvelopeWidths (float vector): Specifies the laser pulse envelope widths. These are defined as twice the standard deviations of the laser pulse intensity envelopes with the first value assigned to the longitudinal envelope length. Required for use with photo-emission Laser pulse.

laserPulseIntensityMagnitude (floating point): Specifies the magnitude of the laser intensity in W/cm2. Required for use with photo-emission Laser pulse.

laserPulsePropagationDirection (float vector): Specifies the direction of propagation of the laser pulse. Make sure that the direction points into the surface of the cathode. By specifying this direction, the incidence angle can be varied. This vector does not have to be specified with a unit length. Required for use with photo-emission Laser pulse.

laserPulseWaveLength (float)

Specifies laser pulse wavelength in m. Required for use with photo-emission Laser pulse.

photonEnergy

Calculated variable to hold hbar*omega for the energy of photons in the laser pulse derived from laserPulseWavelength.

omega

Calculated frequency of laser light derived from laserPulseWavelength.

reflectivity (default = 0.5): Material reflectivity; specifies reflectivity of surface.

scattering: Specifies scattering (F_lambda) for surface material.

fieldEmitterVelGen Example

<VelocityGenerator  emitVelGen>
    kind = fieldEmitterVelGen
    emitVelocityIsLocal = true
    work_function = 4.5
    field_enhancement = 1.0
    multiplier = 1.0
    #
    # Offset for sampling the electric  field
    # alpha is the normalized offset distance where
    # alpha=1 corresponds to the length  of the hypotenuse
    # of one cell. If alpha=1 the electric field is
    # sampled normal to the boundary a  distance alpha*H
    # from the emission point
    # (where H is the hypotenuse). This factor is needed
    # since the field at
    # the boundary is smaller than the field just inside
    # the boundary... The field just inside the boundary
    # is the correct field. Try setting alpha=0 and
    # compare emission currents.
    #
        alpha = 0.1
        temperature = 300.0
    #
    # begin extra parameters needed for txemit emitterType
    #
        emitterType = txemit
        mu = 7.0
        laser = true
    #
    #  below are necessary subparameter is using laser for
    # photo-emission Laser pulse parameters. For now only a
    # pulse with Gaussian envelope shapes in all directions is
    # implemented. The laserPulseStartTime and
    # laserPulseStopTime are the times between which the
    # intensity of the pulse will be calculated at the cathode.
    #
            laserPulseStartTime = 0.0
            laserPulseStopTime = pulseDuration
        #
        # Set the origin from which the pulse is launched. The
        # origin is defined at the pulse maximum value at the
        # startTime.
        #
            laserPulseOrigin = [0.0 $threeLRMS*5.0$ 0.0]
        #
        # Specify the laser pulse envelope widths. These are
        # defined as twice the standard deviations of the laser
        # pulse intensity envelopes with the first value assigned
        # to the longitudinal envelope length.
        #
            laserPulseIntensityEnvelopeWidths = [twoLRMS twoTRMS1 twoTRMS2]
        #
        # Specify the magnitude of the laser intensity in W/cm2.
        #
            laserPulseIntensityMagnitude = 1.0e+9

        #
        # Set the laser pulse wavelength in m.
        #
            laserPulseWavelength = LASER_PULSE_WAVELENGTH
        #


        # Specify the direction of propagation of the  laser pulse.
        # Make sure that the direction points into the surface of
        # the cathode. By specifying this direction, the incidence
        # angle can be varied. This vector does not have to
        # be specified with a unit length.
        #
            laserPulsePropagationDirection = [0.0 1.0 0.0]
     #
     # Specify Reflectivity of surface default set to 0.5
            reflectivity = 0.5

     # Specify Scattering (F_lambda) for surface material
     #
        scattering =  0.043
     #
     # end of extra parameters needed by txemit
     #
        beta_FN = 100.0 #field enhancment for txemit also
     #
        <STFunc  component0>
            kind = expression
            expression = -1.0e6
        </STFunc>
        </VelocityGenerator>