In this chapter, the effect of an external static electric field is described within the BenDa niel-Duke model for the conduction band, introduced in the previous chapter. In Section 4

3. Three Dimensionality
Until now, we have only considered a lD tunneling problem. To calculate charge densities, however, we need to take into account the other two dimensions as well. Here, we use the fact that the energy in the tunneling direction is conserved. Taking z to be the tunneling direction, we can separate in the 3D Density Of States (DOS), , the contributions from the parallel and the perpendicular directions:
(9)
so that we can write for the 3D electronic concentration :

(10)
Here, is the Fermi-Dirac distribution, the moments of which a.re known
as the Fermi~Dirac integrals :

with the inverse temperature. Assuming plane waves in the x- and y-direction, we can express the 2D electronic density in terms of :

with Ne the effective number of states per unit volume in the conduction band (without spin!).
The ID DOS in (10) can be expressed in terms of the wave functions W(z) determined by the transfer matrix approach (see section 2):
(12) Eqs.(10)-(12) allow us to calculate the electronic concentrations at the beginning
of the structure (z=0), at the end and in the well In the z=0 case, the norm squared of the wave function is proportional .to (l+R) (see (8)), which we approximate by 2 since except for ; the remaining integral is the Fermi-Dirac integral of order :
(13)
( takes into account spin degeneracy). In the other two cases and the norm squared of the wave function is proportional to (1-R) (see (8)), which we replace by the -function (6); thus we obtain for the concentration at the end of the structure:

and in the well:
(15)
In (14) and (15), and are to be evaluated at · If we had taken into account more than one resonance, there would have appeared a sum over all resonances in these equations. Since is a rapidly decreasing function of the contribution of higher resonances to the concentrations can in most cases be neglected.
In the same way as we determined the concentrations, we can also determine the current density . Since is independent of z, we can evaluate it at any position. If we concentrate on the end of the structure, we have electrons per unit volume, all with the same velocity component in the z-direction . Using na yields:
(16)
With (16) we have found the current density as a function of . and
The latter quantities should now be expressed in terms of the bias . Before presenting how to do this selfconsistently, we use (15) and (16) to make a second estimate of the dwell time in the well, dividing the (areal) charge density in the well by [7],[12]. It is easily shown that this expression coincides with (7) if , as is the case in a biased DBRTD. A further discussion is postponed to Section 6.