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Fig.4.1 (a) Charge densities in acetimulation and depletion region, relative to the doping concentration , w. the potential drop across the central layers .
(b) Lengths of acetimulation and depletion region, relative to the screening length , vs. .
(c) Potential drops across the acetimtdation region , the central undoped layers , and the depletion region vs. .
(d) Effective Fermi level in accumtdation and depletion region vs.
(4.12)
In Fig. 4.1, the screening lengths and , the charge densities and , the effective Fermi levels for the accumulation and for the depletion layer, and the potential drops across the various layers, obtained from numerically solving (4.6-9), are plotted as functions of the applied bias voltage . It is seen that the linear dependence of (4.12) is quite accurate over a long range of voltages.
The above model for the accumulation and depletion layers is in fact an improved version of the one presented by Joosten et al. 13 In the original version, the depletion layer was left out of consideration. The constituting equations then become:
Here, is to be considered an adjustible parameter. The system (4.13) was developed with an eye to DBRT structures having undoped spacer layers adjacent to the barriers. For such structures Ls is thought to be somehow related to the spacer width, as can be seen from the fact that has no doping concentration dependence. Eq.(4.12), on the other hand, applies to heavily doped electrodes with no spacers.
The function can be determined experimentally from magnetotunneling measurements. When a magnetic field perpendicular to the barriers is applied, oscillations in the current are observed that are periodic in 1/B, with peri dicity . Hence, "measuring" the fundamental field at various biases yields a plot of vs. . The magneto-tunneling results of
Fig.4.2 Fundamental field Br, proportional to the effective Fermi energy, vs. bias voltage . The squares are the measurements of Payling et al.; t he bold curve is the solution of Eqs.(4.6-9); the dotted curve is the solution of (4.13) with nm.
Payling et al are reproduced in Fig. 4.2, together with the theoretical curves of both (4.6-9) and (4.13).
Although the Thomas-Fermi approach (4.4) can be (and has been) exploited in numerically far more sophisticated way than is done in our constant-p model, we nevertheless stick to the latter crude approximation. The fact is, that Thomas-Fermi is unable to deal with the quantum effects that are dominantly present near the barriers. Since the electrons cannot penetrate very far into the barrier, the amplitude of their wave functions will be small near the barrier. Consequently, the electron density is minimal just before the barrier, where Thomas- Fermi predicts a maximum. The Friedel type of oscillations in the density that result from this repellence are of course absent in the semiclassical result. Furthermore, the triangular well in front of the emitter barrier that is formed at finite bias, gives rise to bound states. Hence the accumulation is 2D in character, while the Thomas-Fermi result is a 3D one. Thus one should not consider these Thomas-Fermi densities too realistic. Surprisingly, the potential profiles obtained from the semicla.ssical (4.4) are quite a.curate, especia.lly for the accumulation layer, compared to self-onsistent quantum-mechanical calculations. Our constant-p model (4.6-9) can therefore be motivated thus: for the potential, a crude approximation to (4.4) already suffices, for the charge density, the exact solution of (4.4) still fails.
4.3 Se1fconsistent study of coherent tunneling through a doub1e barrier structure
Abstract - We present a model of the double barrier resonant-tunneling diode (DBRTD), in which the tunneling is described in a 1D transfer matrix approach, based on full wave coherence, and in which the electronic potential is determined selfconsistently from the 3D charge distribution in the structure. Within this simple model, we are able to describe the diode's intrinsic bistability. Results are presented in the form of I-V-characteristics for GaAs-AlGaAs structures. Our approach is evaluated with respect to existing models.
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