Tunneling Transmission Probability: DOS or Velocity?

[asheg]

In some quantum textbooks [1], the tunneling transmission formula depends only on the density of states of 2 regions (DOS) involved in tunneling. (T(E)=C×DOS1(E)×DOS2(E), where C is constant). However, in Landauer transmission formula (without tunneling) the transmission depends on both DOS and velocity of carriers (T(E)=C′×DOS(E)×v). So I am wondering if velocity is important too? If yes, which velocity in which region?!

[1] For example, see "Introduction to Many-body quantum theory in condensed matter physics", Bruus et al.

 

[ZapperZ]

The velocity comes in in the form of the kinetic energy of the tunneling particle. So it is explicitly a factor in any tunneling probability for a finite potential.

Zz.