Space Charge - Child's Law

Child's Law

Also known as the Child-Langmuir Law or the Three-Halves Power Law, Child's Law states that the space-charge limited current (SCLC) in a plane-parallel diode varies directly as the three-halves power of the anode voltage and inversely as the square of the distance separating the cathode and the anode. That is,

.

Where is the anode current, the current density, and the anode surface inner area. This assumes the following:

1. The electrodes are planar, parallel, equipotential surfaces of infinite dimensions.
2. Electrons travel ballistically between electrodes (i.e., no scattering).
3. The electrons have zero velocity at the cathode surface.
4. In the interelectrode region, only electrons are present.
5. The current is space-charge limited.
6. The anode voltage remains constant for a sufficiently long time so that the anode current is steady.

The assumption of no scattering (ballistic transport) is what makes the predictions of Child-Langmuir Law different from those of Mott-Gurney Law. The latter assumes steady-state drift transport and therefore strong scattering.