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Extra info for Electron Emission Physics
If barrier regions are designated by cI and the well regions by cII, Schro¨dinger’s equation is 16 KEVIN L. JENSEN 8 < @2 9 = À 2 Á @ 2 2 þ k u ðxÞ ¼ 0 þ 2ik À k À k o :@x2 ; I @x 8 9 < @2 = À Á @ 2 2 þ k u ðxÞ ¼ 0 þ 2ik À k :@x2 ; II @x ð34Þ hko Þ2 =2m. Solutions are where EðkÞ ¼ ð" hkÞ2 =2m andVo ¼ ð" uI ðxÞ ¼ Aexp½ðkv À ikÞx þ Bexp½Àðkv þ ikÞx uII ðxÞ ¼ Cexp½Àiðk þ kÞx þ Dexp½Àiðk À kÞx ð35Þ where k2v k2o À k2 . Continuity of the wave function implies that uI ð0Þ ¼ uII ð0Þ and periodicity implies that uI ðaÞ ¼ uII ðÀbÞ; these two equations, along with two more relating the first derivatives, provide four equations for four unknown coefficients.
Both allow greater penetration of the emission barrier 45 ELECTRON EMISSION PHYSICS by either a larger population of more energetic electrons (for temperature) or the ability of the electrons to penetrate (for field). Coupled with the dependence of the exchange‐correlation potential on density, as well as the behavior of the dipole contribution, the height of the barrier above the Fermi level changes. It is therefore expected that terms accounting for changes in the work function that depend on temperature, field, and extent of the dipole will need to be introduced, presumably resulting in a change in the definition of F: the image charge potential should therefore resemble V ðxÞ ) mðT Þ þ fðF ; T Þ À Fx À Q ; x þ xo ð115Þ where the use of f accommodates field‐ and temperature‐dependent effects.
93) indicates that ’ðkÞ % À k=ko kxo , so that to leading order the effect of a tall barrier, regardless of width, is to simply shift the density to the left by an amount inversely proportional 1=2 to Vo . The impact of the shift by xo is most readily seen in the zero‐temperature limit for density, which (as for all things quantum mechanical)P is altered from ^ ¼ i ni jiihij in the Eq. (14): the density matrix (Shankar, 1980) is defined as r number representation, where ni ¼ 0 or 1 for fermions.