It is well known that conductivity of insulator increases with electric field because of increased carrier concentration by Poole-Frenkel emission. There is a renowned paper by Murgatroyd dealing with Poole-Frenkel emission with space charge limited current. 

What confuses me is, if Poole-Frenkel emission starts before the onset of space charge, does not it change the onset voltage of space charge because of increased conductivity/concentration? Or in some case, does not it make space-charge effect not to take place at all?  The problem is described below in detail.

In simple understanding, the space-charge starts when the injected charge becomes greater than the thermal equibilibrium charge in material, mathematically,

CV > n0edA ; C is capacitance, V is applied voltage, n0 is concentration, d is thickness and A is area of device.  This can be written as

E>n0ed/epsilon; E is electric field, epsilon is permittivity of material ( I assume permittivity is not much dispersive).

Now suppose that Poole-Frenkel emission starts before space charge. The concentration keeps increasing with the applied field (I don't say that it increases infinitely as it saturates based on trap density). The concentration after including PF is given as

n = n0 exp(\beta\sqrt(E)); beta is PF coefficient and at 300K for a material of dielectric constant 5, it is 0.001319 (m/V)^1/2. 

The condition for space-charge to start becomes

E > (n0 e d/epsilon) exp (\beta\sqrt(E))

 rearranging,

d < (epsilon/n0*e)*E* exp(-\beta\sqrt(E))

This equation, for a material with n0 = 10e14 at 300K, epsilon = 5, gives the maximum thickness of sample around 850 nm.

Suppose we have  a 500 nm thick film. In low field, the field itself is not sufficient to produce space charge. At around 4e3 V/cm, the sample can reach to space charge and reamin in space charge regime up to 8.5e4 V/cm. Beyond that the exponential increase in conductivity (concentration) by PF makes it hard to bring the device in space charge. so the conduction should fall back to Poole-Frenkel only.

The field of 8.5e4 V/cm increases the carrier concentration by 47 times. There are papers on high field saturation of PF effect but the field is not so large, so the number 47 might be possible (unless the trap density is very small and  so not enough carriers are left in trap to be excited by the applied field.)

If this is the case, then how do we get thick devices into space charge regime?

Any help in this is appreciated.

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