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

6component v l , and an electron of the Fermi sea, two electronsemerge with a mean energy ε/2 andameanlongitudinalvelocity component v l /2. We here neglect theenergy and wave vector given to the hole left in the Fermisea by the excitation of the secondary electron. The timeevolution of the mean electron energy and of the meanlongitudinal velocity component can then be written astwo relaxation equations:dεdt = − ln 2 ε τ , (5)anddv ldt = − ln 2v lτ . (6)where τ is the electron-electron collision time. We canthen write a propagation equation of the form :dzdε = dz dtdt dε = −v 1 τlln 2 ε , (7)where v l , the longitudinal component of the mean electronvelocity, is obtained by combining Eqs.(5) and (6):εv l = v 0 . (8)E 0In a parabolic band approximation, the incident electronvelocity v 0 becomes :√E0 + E Fv 0 = v F , (9)E FE F being the Fermi energy. Then, z ball is obtainedafter integration of Eq. (7) :z ball = − 1ln 2∫E ballE 0v l τ dεε = 1ln 2∫E 0E ballλ (ε)√E0 + E Fε + E F(10)where λ (ε) =vτ is the electron mean-free-path. Thevalue of E ball , the mean energy of the electron distributionat the end of the velocity-relaxation transport step,i.e. at the distance z ball from the surface, is obtainedfrom Eq. (8) when taking v l as equal to v F (which isthe criterion chosen for the transition between the twotransport regime) :√EFE ball = E 0 (11)E 0 + E FAfter crossing the distance z ball , we consider that theelectron velocity is relaxed and that the scattering directionis randomized. A three-dimensional diffusion-liketransport regime takes place which can be described bythe evolution equation :dεE 0dz 2dt = 1 D (ε) , (12)3where the quantity D (ε) is similar to an energydependentdiffusion coefficient and is given by :D(ε) =v 2 τ. (13)Along this transport regime, the mean energy of theelectron distribution decreases from E ball to E M accordingto the energy relaxation equation [Eq. (5)]. The propagationequation then becomes:dz 2dε = dz2 dtdt dε = −1 1 v 2 τ 23 ln 2 ε , (14)so that the distance crossed in the diffusion regime isgiven by :zdiff 2 = − 13ln2∫E ME ballλ 2 (ε) dεε . (15)The electron mean energy E M at the junction barrieris then obtained by solving the equation :d = z ball + z diff . (16)For the calculation of z ball and z diff after Eqs.(10)and (15), we have taken E F =7eV , which is close to thevalue of the Fermi energy in palladium, and we have usedfor λ (ε), an empirical variation (plotted in Fig.(9) of theform :( ) al( ) ah Elελ (ε) =λ l + λ h − λ off . (17)ε + E l ε + E hThechoiceofλ (ε) is discussed in Appendix A. We havesolved numerically Eq.(16) and the resulting variation ofE M with injection energy E 0 is plotted in Fig.5 togetherwith the variation of the velocity-relaxation path z ball .In the low injection energy range, i.e. below 80eV, z ballis very short (a few tenth of nanometer) and almost constant: the electron velocity is very quickly relaxed andthe diffusion-like transport step takes place almost allover the metal layer thickness. The result is that E Mremains almost constant and takes a value, small whencompared to both junction barriers φ SC and φ Ox .