Low energy electron conduction in insulators

Abstract

The electrical conduction due to trapped ele£ trom with energy higher than the potential barrier (Poole-Frenkel effect) is claimed to take part in several materials /1/. Much less is known about the current due to electrons whose energy is lower than the trapping well. A classical model is here formulated to take into account such contribution and the theoretical results are compared with tho se experimental ly obtained in CdF 2 :Gd crystals.In these substances, where the electrical con duction is due to a trapping mechanism between lo calized levels, the mean free path of the carriers is determined by the random distribution of the � nized impurity centres. Their density can be equr lized to that of free electrons with energy £ in the conduction band, that is:24 If £�kT with the temperature T=lOOK, one has n�lO m -3 , indicating that the average distance between o charged Gd centres is d � 100 A. On the other hand the de-Broglie wavelength of the electron is: I A = h/((£-V (x» (2m»2 where £�kT and Vex) is the potential wave function: 2 Vex) = e /4rr IxlE ( 2) with � =9G o being the dielectric constant of the o ma terial /2/. Since A<lOO A, the condition required for a classical description is satisfied.134

Internal field-assisted thermally activated hopping and tunnelling in insulators and composite materials

“…Subsequent work on Poole-Frenkel systems indicated the need to account for such fields in order to arrive at a consistent explanation of transport data; the obtained values of the fields agreed with those obtained by optical means [2]. Later on, a theoretical argument based on a diffusion equation was advanced [3] indicating the need for the introduction of an internal field in transport processes. Around coulombic centres, such a field superimposes upon the Coulomb field producing a barrier at zero applied external field, and hence in the ohmic region, which the carriers can overcome either assisted by the temperature or by tunnelling at low temperatures.…”

“…In this case the internal field L is solenoidal around the top of the barrier, as follows from equation ( 4), implying absence of polarization charge. Despite the particular configuration of the fields E and L, equation ( 8) has been shown to be a notable improvement of the original Poole-Frenkel result (corresponding to L = 0) and more extensive analysis including realistic configurations [2,3] has further shown the relevance of the internal field corrections.…”

“…to a polarization charge. The internal field corrections have been previously discussed [2,3] in Poole-Frenkel systems. We review such a case to introduce the results of the other insulators where conduction occurs at the Fermi surface.…”

“…a saddle-point method, looking at stationary points of f P (ξ ). If r s represents such a point we can assume then σ ≈ f (r s )P (r s ), for constant n and µ [2][3][4]. When the distribution P (ξ) is exponential and f is given by (3), stationarity leads to the condition for r s…”

“…The charge transport in a disordered system can be studied with the Fokker-Planck equation in the Smoluchowski limit for the distribution function f [3]:…”