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

4.Selfconsistency

Given V1and V2 we are able to determine the resonance levels, and and at these energies. Given, in addition, the Fermi level , we a.re able to evaluate the electronic concentrations in the structure and the current density. In reality, the device is part of a.n electric circuit: a bias voltage is applied to it, a.nd the current is measured. So what we need in order to determine the I-V curve, are the functions that relate and to . Because we also want to include the phenomenon of bistability, we will not look for etc., for these will not be single-valued functions. In stead, we will .option for a parametric description, using as a parameter, a.nd determine the functions and

The central relation of our model then reads:

(17)
It states that both the filling of the left-hand reservoir (characterized by ) and the bending of the conduction band edge (expressed by and ) result from the application of . The band bending is due to accumulation up charge in the central, undoped layers of the DBRTD. Therefore, and can be related to the charge concentrations at the beginning of the structure and in the well:
(18a)
(18b)
Eq.(18) is nothing but a simplified Poisson equation: in fact, we have assumed the charge to be concentrated in two narrow sheets, one positioned at z = 0, the other at , and calculated the potential drops in capacitor analogy. Substituting in (18) the results (13) and (15), we find:
(19a)
(19b)
This would conclude our task, since (19) suggests that and are found. However, depends on the resonance energy, and, in turn, to determine Eres we need to know . This problem is another aspect of selfconsistency that every model of the DBRTD encounters. In our simple model it only involves the potential drop across the second barrier. More generally, selfconsistency is the demand to solve the SchrOdinger equation and Poisson's equation simultaneously. Before presenting our solution of the set of equations (16), (17) and (19) (see Section 5), we will introduce some simplifications, giving insight into the nature of the bistability and allowing us to make a direct comparison with literature,



Fig. 2 The I- V characteristics resulting from the simple approximation (20) at zero temperature; the parameter values used are those of the symmetric sample of Section 5.

especially with Sheard and Toombs [7]. Following [7], let us assume the resonance energy to be constant with respect to the bottom 0£ the well ; this is a very reasonable approximation as can be seen from Fig.3a, where vs. is drawn. (as calculated with the exact equations (16)-(19)). Let us further assume zero temperature. We define the constants A,B and C: