Pierce Diode at the Child–Langmuir Limit

This example shows how to simulate the physics of a 1D Pierce diode configuration operating at the Child–Langmuir limit using WarpX. In this setup, an electron beam is injected into a planar diode gap, consisting of by parallel conducting plates separated by the distance \(d\) and powered by a voltage difference \(V\) Fig. 12. The injected current density is chosen to match the space-charge-limited current predicted by the Child–Langmuir law Zhang et al. [16]. The law predicts the maximum current density that can flow between two parallel plates due to space-charge effects. This test demonstrates that WarpX correctly reproduces the Child–Langmuir law for a given voltage and gap length.

Geometry

The figure below schematically illustrates the problem geometry described above.

Fig. 12 Two parallel conducting plates separated by the distance \(d\) and powered by a voltage difference \(V\). Given that the two plates are parallel, here we simulate the problem in 1D with WarpX.

Сhild–Langmuir Limit

In steady state, the emitted current is limited by the Child–Langmuir law, which defines the maximum current that can be transported across a planar diode for a given voltage and gap length Zhang et al. [16]. It can be shown that, at the Child-Langmuir limit (i.e. when this maximum current is reached), the potential and current density in the gap have the following expression:

(1)\[\phi(z)=V\Big(\frac{z}{d}\Big)^{4/3},\]

(2)\[J(z) = \frac{4}{9} \varepsilon_0 \sqrt{\frac{2 |q|}{m}} \frac{|V|^{3/2}}{d^2}.\]

Run

This example can be run with the WarpX executable using an input file: warpx.1d inputs_test_1d_pierce_diode. For MPI-parallel runs, prefix these lines with mpiexec -n 4 ... or srun -n 4 ..., depending on the system.

Listing 65 You can copy this file from Examples/Physics_applications/pierce_diode/inputs_test_1d_pierce_diode.

# --- This input defines 1D Pierce Diode Example:
# --- This test simulates a classic Pierce diode configuration:
# --- two parallel conducting plates separated by a distance
# --- d_plate = 8 cm and powered by a voltage difference
# --- extractor_voltage = -93 kV. The injected current density is
# --- chosen to match the space-charge-limited current predicted by
# --- the Child–Langmuir law, that predicts the maximum current density
# --- that can flow between two parallel plates due to space-charge effects
# --- for a given voltage and gap length.

# constants
my_constants.ion_mass = 39*m_u # [kg] Assuming ion potassium
my_constants.kV = 1000
my_constants.cm = 0.01
my_constants.extractor_voltage  = -93.*kV
my_constants.d_plate = 8.*cm
my_constants.nz = 128
my_constants.dz = d_plate/nz
my_constants.vzfinal = sqrt(2.*abs(extractor_voltage)*q_e/ion_mass)
my_constants.dt = 0.4*(dz/vzfinal)
my_constants.ep0 = 8.8541878188e-12
# current density at Child-Langmuir limit
my_constants.J_CL = ( (4 / 9)* ep0 * sqrt(2 * abs(q_e) / ion_mass) * abs(extractor_voltage) ** (3 / 2) / d_plate**2 )
# algo
warpx.do_electrostatic = labframe
algo.particle_shape = 1

# amr
amr.n_cell = nz
amr.max_level = 0

# maxamx step
max_step = 5000

# timestep
warpx.const_dt = dt

# number of procs
warpx.numprocs = 2   # 2 MPI ranks

# geometry
geometry.dims = 1
geometry.prob_lo = 0.0
geometry.prob_hi = d_plate

# boundary
boundary.field_lo = pec
boundary.field_hi = pec
# set fixed potential at the plates (V difference)
boundary.potential_lo_z = 0.0      # cathode at x = 0 V
boundary.potential_hi_z = extractor_voltage  # anode at x = 1 kV
boundary.particle_lo = absorbing
boundary.particle_hi = absorbing

# ions
particles.species_names = ions
ions.charge = q_e
ions.mass = ion_mass
ions.do_continuous_injection = 1
ions.injection_style = "NFluxPerCell"
ions.num_particles_per_cell = 15
ions.surface_flux_pos = 0
ions.flux_normal_axis = z
ions.flux_direction = 1
ions.flux_profile = constant
ions.flux = J_CL/q_e # corresponds to Child-Langmuir limit
ions.momentum_distribution_type = constant

# diagnostics
diagnostics.diags_names = diag1
diag1.intervals = 5000
diag1.diag_type = Full
diag1.format=openpmd
diag1.fields_to_plot = Ez rho jz phi

Visualize

The figure below shows the results of the simulation (orange curves), which agrees well with the analytical Child–Langmuir law (black curves) (1, 2).

This figure was obtained with the script below, which can be run with python3 plot_sim.py.

Listing 66 You can copy this file from Examples/Physics_applications/pierce_diode/plot_sim.py.

import matplotlib.pyplot as plt
import numpy as np
from openpmd_viewer import OpenPMDTimeSeries
from scipy.constants import c, e, epsilon_0, m_u

ts = OpenPMDTimeSeries("./diags/diag1/")

kV = 1000
cm = 0.01
extractor_voltage = -93.0 * kV
d_plate = 8.0 * cm
ion_mass = 39 * m_u

it = ts.iterations[-1]
phi, meta = ts.get_field("phi", iteration=it, plot=False)
ez, _ = ts.get_field("E", "z", iteration=ts.iterations[-1], plot=False)
rho, _ = ts.get_field("rho", iteration=it, plot=False)
jz, _ = ts.get_field("j", "z", iteration=it, plot=False)
z, uz = ts.get_particle(["z", "uz"], iteration=it)
time_cur = ts.current_t

# Calculate theoretical Child-Langmuir limit for a given voltage
jz_CL_theory = (
    (4 / 9)
    * epsilon_0
    * np.sqrt(2 * abs(e) / ion_mass)
    * abs(extractor_voltage) ** (3 / 2)
    / d_plate**2
)
phi_CL_theory = extractor_voltage * (meta.z / d_plate) ** (4 / 3)
ez_CL_theory = -(4 / (3 * d_plate)) * extractor_voltage * (meta.z / d_plate) ** (1 / 3)
rho_CL_theory = (
    epsilon_0
    * (4 / (3 * d_plate) ** 2)
    * extractor_voltage
    * (meta.z / d_plate) ** (-2 / 3)
)
uz_CL_theory = -jz_CL_theory / rho_CL_theory / c

color = "orange"
title = r"$\Gamma_{ions}=38.79 \approx \Gamma_{CL}$"

fig, axs = plt.subplots(2, 2, figsize=(10, 8))
fig.suptitle(f"{title}\n time = {np.round(time_cur / 1e-6, 5)} $\\mu s$")
axs[0, 0].scatter(z, uz, color=color, label=title, s=0.2)
axs[0, 0].plot(meta.z, uz_CL_theory, ls=":", color="black")
axs[0, 0].set_title(r"$u_z$")
axs[0, 0].set_xlabel("z, mm")

axs[0, 1].plot(meta.z, ez, color=color, label="WarpX ")
axs[0, 1].set_title(r"$E_z$")
axs[0, 1].set_xlabel("z, mm")
axs[0, 1].plot(
    meta.z, ez_CL_theory, label="Child-Langmuir limit (theory)", ls=":", color="black"
)
axs[0, 1].legend()

axs[1, 0].plot(meta.z, jz, color=color)
axs[1, 0].set_title(r"$J_z$")
axs[1, 0].axhline(y=jz_CL_theory, color="black", linestyle="-")
axs[1, 0].set_xlabel("z, mm")

axs[1, 1].plot(meta.z, phi, color=color)
axs[1, 1].set_title(r"$\phi$")
axs[1, 1].set_xlabel("z, mm")
axs[1, 1].plot(meta.z, phi_CL_theory, label="theory", ls=":", color="black")

plt.tight_layout()

plt.savefig("Pierce_Diode.png")