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

13ergies of the total electron distribution if the metal wasnon-magnetic and if the electron mean-free-path variationwith ε was equal to λ + (ε) andλ − (ε), respectively.The variations λ + (ε) andλ − (ε) thatwehaveusedaredefined as usually by:1λ ± (ε) ≈ 1 [1 ∓ λ (ε) ]. (B1)λ (ε) δ (ε)FIG. 9: Variation of the electron mean-free-path λ (ε) versusenergy above the Fermi level. This curve is plotted after theempirical analytical form of Eq.(16) using the following set ofparameters:λ l =14.5nm,λ h = 11nm,λ off =4.8nm,E l =0.5eV,E h = 1000eV, a l =0.35,a h =0.5 (see Appendix A). The insetpresents a zoom in the region of the minimum of λ (ε) togetherwith measured values (taken from Ref. 5 )ofλ + and λ − themean-free-paths of majority- and minority-spin electrons iniron.where λ (ε) is the mean-free-path as defined inSec.III.A.2 and in Appendix A, and δ (ε) is the socalledspin-discriminating length. 14,18 The ratio λ/δ =(λ + − λ − ) / (λ + + λ − ) is the spin asymmetry of the electronmean-free-path. The measured values of λ/δ in irondeduced from Ref.5 are plotted in Fig.10 (square symbols).It decreases linearly with energy. So, we use in Eq.B1the following variation:i) λ/δ =0.15 (1 − ε/50), for ε50eV .The variation of E + M and E− Mthat we obtain afterthis calculation verify reasonably well the relationE + M + E− M =2E M , as shown in Fig.11. These values arethose used in the calculation of ΔT plotted in Fig.6(b)of the main text. Using Eqs.(29-30) of the main text andAPPENDIX B: SPIN-ASYMMETRY OF THETRANSMITTED ELECTRON DISTRIBUTION.The evaluation of the spin asymmetry of the primaryelectron mean energy A Ep andoftheprimaryelectrondistribution A p (ε), according to Eqs.(27), (31) and (32)of Section III.B, requires the calculation of E + M = E+ p /2and E − M = E− p /2, E p + and Ep − being the mean energiesof respectively the majority-spin and minority-spin primaryelectrons. For this calculation, we use basically thesame procedure as for the calculation of E M the electronmean energy at the metal / oxide interface (seeSec.III.A.2 and Appendix A), except that we take intoaccount the spin-asymmetry of the electron mean-freepath.We consider that spin-dependent collisions onlyoccur during the diffusion-like transport regime (indeedthe velocity-relaxation transport step all takes place inthe Pd layer, as shown in Fig. (5) of Sec.III.A.2). We thencalculate z + diff and z− difffor majority-spin electrons andfor minority-spin electrons by using two different variationswith energy of the electron mean-free-path λ + (ε)and λ − (ε), respectively. Then, solving the equationd = z ball + z ± diffgives two different values of the electronmean energy: E + M and E− M. It is important to notethat E + M and E− Mare actually not the mean energies ofmajority- and minority-spin electrons but the mean en-FIG. 10: Variation of λ/δ, the spin-asymmetry of the electronmean-free-path in iron deduced the values of λ + and λ −(inset) reported in Ref.5. The dashed line corresponds to thevariation of λ/δ that we have used in the expression of λ +and λ − of Eq.(B1) for the calculation of E + M and E− M .assuming that f s (ε) ≈ f(ε) andE s ≈ E M ,wemaywriteΔT on the following convenient form:ΔT ≈ 2P 0∫0∞α (ε)[ εE MA p (ε) − 2A Ep]f (ε) dε.(B2)The spin asymmetry of the primary electron distributionA p (ε) determines the spin filtering effect and thespin-asymmetry of the primary electron mean energyA Ep determines the spin-dependent secondary electron