Child–Langmuir Space-Charge Diode

The Child–Langmuir law describes the space-charge-limited current in a planar vacuum diode. Unlike a semiconductor Shockley junction where current follows an exponential relation, a vacuum diode with plentiful electron emission from a cold cathode has its current limited by the electrostatic repulsion of the electrons already in the gap.

For a cathode at potential 0 and an anode at potential V > 0 separated by a gap d, the magnitude of the space-charge-limited current density is J = (4/9)ε₀√(2e/mₑ) V^(3/2) / d². The potential profile is φ(x) = V (x/d)^(4/3), yielding an electric field E(x) = -(4V/3d) (x/d)^(1/3) and electron speed v(x) = √(2eV/mₑ) (x/d)^(2/3). The negative charge density is ρ(x) = -J / v(x). At the cathode (x = 0), the density diverges, which reflects the singular limit of the macroscopic fluid model assuming zero initial velocity.

Worked example: Current and fields

Consider a planar vacuum diode with a gap d = 2 mm (0.002 m) and an applied voltage V = 100 V. Let the position over the gap be halfway (x/d = 0.5). Using fundamental constants ε₀ ≈ 8.854e-12 F/m, e ≈ 1.602e-19 C, and mₑ ≈ 9.109e-31 kg:

J = (4/9) * 8.8541878128e-12 * sqrt(2 * 1.602176634e-19 / 9.1093837139e-31) * (100)^(3/2) / (0.002)^2 = 583.4879842199841 A/m² φ = 100 * (0.5)^(4/3) = 39.68502629920499 V E = -(4 * 100) / (3 * 0.002) * (0.5)^(1/3) = -52913.36839893999 V/m v = sqrt(2 * 1.602176634e-19 * 100 / 9.1093837139e-31) * (0.5)^(2/3) = 3736276.710524766 m/s ρ = -583.4879842199841 / 3736276.710524766 = -0.00015616830053736367 C/m³

Experiment with voltage, gap size, and position

Open the Child–Langmuir Space-Charge Diode module. Interact with the controls to see how space charge affects potential curvature and current density scaling.

Set Voltage V to 100, Gap millimeters to 2, and Position over gap to 0.5. Verify the current density is 583.4879842199841 A/m² and the potential is 39.68502629920499 V.
Set Voltage V to 1000. Notice the current density increases to 18451.510174755356 A/m² and the potential is 396.8502629920499 V.
Set Voltage V to 100, Gap millimeters to 1, and Position over gap to 0.5. Verify the current density is 2333.9519368799365 A/m². The smaller gap increases the field and thus the current.
Set Gap millimeters back to 2, and change Position over gap to 0. Observe the charge density becomes "Unavailable" as it diverges at the cathode.
Set Position over gap to 1. Verify the potential is exactly 100 V, the anode voltage.

Ideal-model limitations

The Child–Langmuir model assumes a purely 1D planar geometry, infinite parallel plates, perfectly uniform plentiful electron emission, zero initial emission velocity, and no collisions. Relativistic effects are neglected. In a real diode, initial thermal velocities round off the divergence at the cathode, forming a virtual cathode slightly ahead of it. The model does not apply to semiconductor junctions or complex non-planar geometries without modification.

References

For an in-depth derivation and discussion of the space-charge-limited current and the Child–Langmuir law, see: Physics of Plasmas - Space-charge-limited current.

Child–Langmuir Space-Charge Diode