Estimation of the Space-Charge Limited Current

We note that the typical dependence results of the voltage dependent charge density, which contrasts with the ohmic conduction where the bulk carrier density is independent of the applied voltage. The thickness dependence in L results of the L dependence of the injected charge density, as opposed to the ohmic carriers density, which is independent of the thickness. 

The derivation of the exact expression of the space charge limited current would require the resolution of Eq. (27) in the case of a non zero current. Unfortunately, such a resolution cannot be performed analytically. The standard expression of the space charge limited current (Eq. 30), as derived by Lampert [49], was obtained by neglecting the diffusion current (second term of the right hand side of Eq. (26)). It is in fact very close to that deduced from the phenomenological model developed above. 

As the thickness L of the insulating film decreases, the concentration of injected carrier increases, and may become important when L is lower that Xq. If the minimum concentration at x = L/2 exceeds that of the extrinsic carriers, the linear current will only results from injected charges. This is likely to occur when L Xq. Such a space-charge limited linear current can be distinguished from the conventional ohmic current from its thickness and temperature dependence. We shall come back to this linear regime of the SCLC in the next section. 

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