We study the combined effect of a Laplacian field and quenched disorde
r in the generation of fractal structures in the quenched dielectric b
reakdown model. The growth dynamics is shown to evolve from the avalan
ches of invasion percolation (IF) to the smooth growth of Laplacian fr
actals, i.e., diffusion limited aggregation and the dielectric breakdo
wn model (DBM). The fractal dimension is strongly reduced with respect
to both the DBM and IF, due to the combined effect of memory and fiel
d screening. This implies a specific relation between the fractal dime
nsion of the breakdown structures and the microscopic properties of di
sordered materials.