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

12APPENDIX A: VARIATION OF THE ELECTRONMEAN-FREE-PATH WITH ENERGY.The model developed for the calculation of the transmissioninvolves many parameters. In order to obtaina reliable fit of the data with a reasonable number ofadjustable parameters, we have used a data fitting procedurein two steps.ThefirststepisthedeterminationofthevariationofE M with E 0 , from the measured values of T and usingEq.(21), we have deduced. Second, from Eqs.(10), (15)and (16) of our energy and velocity relaxation model,we have calculated the variation of E M with E 0 . Wehave fitted the experimental variation of E M with thecalculated one by adjusting λ(ε) the variation of theelectron-mean-free path with energy. he experimentalvariation of E M can be obtained from the experimentaldata of the transmission T by solving Eq.(21). Thisrequires knowing the two barrier heights and the valuesof the transfer efficiency above these two barriers. Forthe barrier heights, we have used the measured valuesφ SC =0.78eV , φ Ox =4.5eV . 24−26 For the transmissioncoefficients we have taken α SC =10 −4 ,andα Ox =0.5.We have first chosen arbitrarily the value of α Ox =0.5that we estimate to be reasonable because the high limitof α Ox is 1 and it can hardly be very much smaller thanunity since the overall electron transmission reaches valuesmuch larger than unity. We have then tried differentvalues for α SC and we have found that α SC =10 −4 yieldsthe most reasonable variation of E M deduced from thetransmission data (square symbols in Fig.8). Indeed, thevariation of E M obtained with the above set of parametersis rather smooth. This is a good criterion for selectingreliable values of the transfer efficiency. Indeed, whenchanging the value of α SC we obtain variations of E Mwhich exhibit unphysical features as shown in Fig.8 (thedotted lines correspond to the values α SC =0.5 × 10 −4and α SC =2×10 −4 ). Therefore, with this criterion, onlyone of the two transmission coefficients is indeed an adjustableparameter, the other one being determined bythe shape of the variation of E M with E 0 .The second step of the calculation consists in fitting the“experimental” variation of E M versus E M versus E 0with the theoretical variation calculated from the modeldescribed in Sec.III.A.2. In this calculation, we havetaken E F = 7eV which corresponds to the Fermi energyin palladium, and we have adjusted the variation ofthe electron mean-free-path λ(ε). We have first chosenfor λ (ε) an empirical form:λ (ε) =λ l() al(El+ λ hε + E l) ahε− λ off .ε + E h(A1)This variation reproduces the main features of the wellknownuniversal curve. 27 The first term in Eq.(A1) givesthe decrease of λ(ε) in the low energy range while thesecond term gives the high energy increase of λ(ε). Thethird term λ off is a constant that we use to adjust theFIG. 8: Experimental (symbols) and calculated (full line)variation with E 0 of the electron mean energy E M at themetal/oxide interface. The experimental variation is obtainedfrom the transmission data by solving Eq.(21) with α SC =10 −4 and α Ox =0.5. When using in Eq.(21) α SC =0.5×10 −4or α SC =2× 10 −4 instead of α SC =10 −4 (dotted lines) theexperimental variation of E M exhibits unphysical features inthe vicinity of the transition between the two transmissionregime, i.e. between 200eV and 500eV injection energy.minimum value of λ(ε). Indeed, it is known that λ(ε)reaches a minimum value λ min of the order of a few tenthof nanometers at an energy E min of several tens of eV.Moreover, the subsequent increase of λ(ε) towards highenergy is proportional to the square root of ε. Therefore,we can impose three conditions to the shape of λ(ε):a h =0.5,λ min =0.5nm, E min =40eV(A2)The above values of λ min and E min are typical valuesthat we have chosen arbitrarily. The three above conditionsallow to reduce the number of independently adjustableparameters from seven [Eq.(A1)] to four, whichis not that much when considering that we are probingthe transport over an energy range which goes fromabout 0.1eV up to 1keV and that we are actually makingmeasurements over almost this entire energy range.The fit of the variation of E M versus E 0 that we finallyretained is plotted in Fig.7(full line). It is obtainedfor the following set of parameters: 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. The variation of λ (ε) associatedthis set of parameters is plotted in Fig.9.It is in agreement with the shape of the universal curveand with the measurements of the mean-free-path in Fetaken from Ref. 5 and reproduced in Fig.9.Note that the obtained value of a l =0.35 is smallerthan the value predicted by several models describingthe universal curve. However, moderate increase in theelectron mean-free-path is often reported specifically intransition metals. This is for instance the case of the experimentalmeasurements taken from Ref. 5 that we haveused to estimate the spin-asymmetry of the mean-freepathin iron.