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

9Let us briefly consider the case of an unpolarized incidentelectron beam injected in the magnetic metal layer.We can define four quantities which characterize the primaryelectron distribution at the metal/oxide interfaceand which are useful to treat the case of a polarizedincident electron beam. First, the total primary electrondistribution f p (ε) is the sum of the majority- andminority-spin primary electron contributions:f p (ε) = f p + (ε)+fp − (ε). (24)2Second, we note A p (ε) the spin asymmetry of f p (ε),which also represents the energy distribution of the primaryelectron polarization that is generated by the spinfiltering effect:A p (ε) = f + p (ε) − f − p (ε)f + p (ε)+f − p (ε) . (25)Then, we note E p the mean energy of the primary electrondistribution f p (ε):E p = E+ p + Ep− (26)2and A Ep the spin asymmetry of the primary electronmean energy:(24)A Ep = E+ p − Ep−E p+ + Ep− . (27)Then, using Eq.(23-27) in Eq.(22), we can writeF (ε, ±P 0 ) in a convenient form:F (ε, ±P 0 )= f p (ε)+(M − 1) f s (ε) (28)[]E p±P 0 A p (ε) f p (ε) − A Ep f s (ε)E swhere M =(M + + M − )/2.The terms f p (ε)+(M − 1) f s (ε) in Eq.(28) correspondto the electron distribution at the junction when an unpolarizedelectron beam is injected into the metal layer.In the following, we will compare these two terms withthe electron distribution F (ε) as defined by Eqs.(1), (4)and (20) of the previous section: f p (ε)+(M − 1) f s (ε) =F (ε) =Mf (ε) =(E 0 /E M )(1/E M )exp(−ε/E M ). Now,if we consider that the barrier transmission coefficientα (ε) does not depend on spin, the transmission T (±P 0 )for the two opposite values of the incident polarizationis obtained by integration of Eq.(28) and the spindependenttransmission ΔT = T (+P 0 )−T (−P 0 )isgivenby:∫∞[]E pΔT =2P 0 α (ε) A p (ε) f p (ε) − A Ep f s (ε) dε.E s0(29)The model that we have developed for describing theenergy relaxation by excitation of a secondary electroncascade implies that, at a given energy ε, theratioofthe total number of electrons to the number of primaryelectrons is simply given by E 0 /ε: this expresses that theenergy E 0 of the primary electron is shared by energy E 0of the primary electron is shared by E 0 /ε electrons ofenergy ε. 28 Therefore, the primary electron distributionf p (ε) can be simply obtained from the expression of theoverall electron distribution F (ε):f p (ε) = ε F (ε) = ε (1exp − ε )(30)E 0 E M E M E MNote that, the mean energy of the primary electrondistribution f p (ε) as defined by Eq.(28) is twice the energyof the total distribution f (ε): E p =2E M . Therefore,we can take E p /E s ≈ 2 in Eq.(29) since M is muchlarger than unity in the whole explored energy range.The above description of the primary electron distribution,does not allow to determine separately f p+ (ε) andfp− (ε). Therefore, we will take an empirical approximationfor A p (ε):A p (ε) ≈ g+ (ε) − g − (ε)g + (ε)+g − (ε) , (31)where g + (ε) andg − (ε) are defined by:g ± (ε) =ε (1E ± ME ± exp − ε )ME ± . (32)MWe calculate E + M and E− Mfollowing the energy relaxationmodel of Section III.A.2. For these two calculations,we use two variations of the electron mean-freepathwith energy, respectively λ + (ε) andλ − (ε), whichare obtained from λ (ε) by introducing the spin asymmetryof the electron mean-free-path deduced from Ref( 5 )for details, see Appendix B). With the values of E + M andE − Mobtained this way, we evaluate both the primary electrondistribution asymmetry A p (ε) [from Eqs.(31) and(32)] and the primary electron mean energy asymmetryA Ep [from Eq.(27)]. We finally obtain from Eq.(29) thevariation of ΔT plotted in Fig.6(b). This calculation,for which we have not used any other adjustable parameters,is in very good agreement with the experimentaldata [square symbols in Fig.6(b)]. We have consideredthe same step shape for the junction collection efficiencyα(ε) as in Sec.III.A. The two calculated contributionsof the electrons transmitted above the barrier φ SC andabove the barrier φ Ox are plotted separately [dotted linesin Fig.6(b)]. As it was already demonstrated on the variationof T , the stiff raise in ΔT above 350eV injection