Etudes par microscopie en champ proche des phénomènes de ...

7n = −∫E ME 01ln 2dεε = 1ln 2 ln (E0E M). (19)Since each collision yields two electrons (the incomingelectron and the secondary electron excited from theFermi sea), the multiplication factor M is:FIG. 5: Calculated variation of the electron mean energy E Mat the metal oxide interface and of the distance z ball crossedthrough the metal layer as a function of the injection energyE 0.In the high injection energy range, i.e. above typically80eV, z ball starts to rapidly increase and reachesa value of several nanometer at 1000eV, which is a significantpart of the total metal layer thickness. Velocityrelaxation requires a longer path, so that the electronspenetrate more deeply into the metallic layer before thediffusive regime takes place. Therefore, energy relaxationis less efficient and E M increases. It even reaches valueslarger than the semiconductor band bending φ SC .3. Calculation of T as a function of E 0The average number n of collisions that an electronundergoes during the transport through the metal layeris given by:n =∫t M0dtτ(18)where t M is the total time that takes the average electronto cross the metal layer. During this time, the averageelectron cascades from E 0 , the injection energy, toE M , the mean energy at the metal/oxide interface. Accordingto Eq.(5), the average number of collisions duringthe transport is given by:M =2 n = E 0. (20)E MThe multiplication factor simply reflects the fact thatthe primary electron energy E 0 is shared with the M electrons(the primary and the secondaries) of mean energyE M .Combining Eqs.(3), (4) and (20), and assuming thatE 0 >> E M and α Ox >> α SC , we then obtain a simpleexpression for the transmission through the junction:T ≈ E [ (0α SC exp − φ ) (SC+ α Ox exp − φ )]Ox.E M E M E M(21)Using the variation of E M with E 0 calculated in theprevious section, we obtain the theoretical variation ofthe transmission T versus E 0 plotted in Fig.6(a).For this calculation, we have used the values of the twobarrier heights already mentioned: φ SC =0.78eV andφ Ox =4.5eV , and we have taken for the transmissioncoefficients α SC =10 −4 and α Ox =0.5 whicharereasonableestimations of the transmission probability aboveφ SC (through the oxide barrier) and above φ Ox , respectively.Details concerning the choice of the parametersand the fitting procedure are given in Appendix A. Thecalculated variation of T reproduces the three regimesobserved experimentally. In the first regime, the linearincrease of T with E 0 , is due to the multiplication factorM = E 0 /E M because the electron mean energy E Mremains almost constant as shown in Fig.5. For injectionenergies larger than 80eV, the second regime starts,where T increases faster than linearly. As previouslymentioned, due to the increase in the injected electron velocityand to the increase in the electron mean free path,the velocity-relaxation path z ball increases, which causesan increase in the electron distribution mean energy E Mat the metal oxide interface. Thus a larger number ofelectrons may overcome the barrier and be transmittedin the semiconductor. In the third regime, over 350eV injectionenergy, the electron mean energy has increased sothat the transmission is dominated by electrons of energyhigher than the oxide barrier height φ Ox . The two contributionsto the transmission above φ SC , through theoxide barrier, and above φ Ox are plotted separately inFig.6(a) (dotted and dashed lines). It is clear that withthe increase in E M the transmission goes from a regimedominated by electrons of energy just larger than φ SC(despite the fact that α SC is very small when comparedto α Ox ) to a regime dominated by electrons of energylarger than φ Ox .