Equations of the Model
The device under consideration is based on a GL or MGL structures with p- and n-side sections with the contacts and an undoped transit i-section. For definiteness, we assume that p- and n-sections in a GL or in an MGL are created by chemical doping [13], although in similar devices but with extra gates, these sections can be formed electrically [10, 11]. Under the reverse bias voltage V0, the potential drops primarily across the i-section A schematic view of the structures in question and their band-diagram under sufficiently strong reverse bias (V0 > 0) corresponding to the potential distribution in the i-section with a markedly nonuniform electric field are shown in Fig. 10.1.
The electrons and holes injected into the i-section of one GL are characterized by the electron and hole sheet concentrations, E" = Z"(t, x) and E+ = E+(t, x), respectively, and the potential tp= tp (t, x, y). Here, the axis x is directed along the GL (or MGL) plane, i.e., in the direction of the current, the axis у is directed perpendicular to this plane, and t is the time. In the general case when the i-section is based on MGL, I" = E"(t, x) and I+ = L+(t, x) are the electron and hole densities in each GLs. In the case of BET and BHT, on which we focus mainly, the electron and hole sheet densities in each GL obey the continuity equations
and the Poisson equation
respectively. Here, e =|e| is the electron charge, ae is the dielectric constant of the media surrounding GL (or MGL structure), К is the number of GLs in the GTUNNETT structure, and <5(y) is the delta function reflecting a narrowiness of GL and MGL structures even with rather large number of GLs in the у-direction. Equation (10.1) corresponds to the situation when the electrons and holes generated due to the interband tunneling in the i-section obtain the directed velocities vx = щ and u* = -Uw and preserve them
Figure 10.1 Schematic views of GTUNNETT p-i-n diodes (a) with a single GL, (b) with an MGL structure, and (c) their band diagram at reverse bias. Arrows show the propagation directions of electrons and holes generated due to interband tunneling (mainly in those regions, where the electric field is relatively strong).
during the propagation. The boundary conditions correspond to the assumption that the electrons and holes appear in the i-section only due to the interband tunneling (the injection of electrons from the p-section and holes from the n-section is negligible) and that the highly conducting side contacts to the p- and n-sections are of blade type (the thicknesses of GL and MGL and the contacts to them are much smaller than the spacing between the contacts)
where 21 is the length of the i-section and V = V0 + SVW exp(-/cot) is the net voltage, which comprises the bias voltage 70 and the signal component with the amplitude 57 and the frequency со. The interband tunneling generation rate of electrons and holes in each GL (per unit of its area) is given by [8-11]
where
so that the characteristic tunneling dc current (per unit length in the transverse z-direction) and the characteristic electron and hole sheet density can be presented as
From Eq. (10.2) with boundary condition (10.4), for the potential (p(x) in the GL (or MGL structure) plane, we obtain
where
Introducing the dimensionless quantities: «+ = £ = x/l,
= x'/I, and т = ivt//, Eqs. (10.1) and (10.6) can be reduced to the following system of nonlinear integro-differential equations:
Here,
The term with the factor yin the right-hand side of Eq. (10.8) is associated with the contributions of the electron and hole charges to the self-consistent electric field in the i-section.
The dc and ac components of the terminal dc and ac currents, J0 and (5/(!> are expressed via the dc and ac components rig and 8nf0 as follows:
Here, p[$) = l/7CyJl-%2 is the form factor and C ~ ae/2^2 is the geometrical capacitance [14,15]. The explicit coordinate dependence of the form factor is a consequence of the Shocley-Ramo theorem [16,17] for the device geometry under consideration.
Hence the GTUNNETT small-signal admittance Ya = 8]J8Va is presented in the form
As follows from Eq. (10.8), the problem under consideration is characterized by the parameter у
If ae = 1.0, K= 1, and 21 = 0.7 pm, in the voltage range V0 = 100 - 200 mV, one obtains £0 (0.6 - 1.7) x Ю10 cm"2 and y^ 0.61 - 0.87.
Spatial Potential Distributions and Current-Voltage Characteristics
If the charges created by the propagating electrons and holes in the i-section are insignificant, that corresponds to у<к 1. In this case, the potential distribution is given by
and Eqs. (10.8), neglecting the term with у can be solved analytically. Takinginto account boundary conditions Eq. (10.3), from Eqs. (10.8), we obtain
After that, using Eq. (10.11) and considering Eq. (10.15), one can find the following formula for the dc current-voltage characteristic:
where Г(х) is the Gamma-function. A distinction between J0 and Jqq is due to the nonuniformity of the electric field in the i-section associated with the feature of the device geometry taken into account calculating J0. At К = 1, 21 = 0.7 pm, and V0 = 100 - 200 mV, Eq. (10.16) yields J0 ^ 0.18 - 0.51 A/cm.
To take the effect of the self-consistent electric field on the tunneling, one needs to solve system of Eqs. (10.8). Due to a complexity of the nonlinear integro-differential equations in question, a numerical approach is indispensable. Equations (10.8) were solved numerically using successive approximation method, which is valid when y< yc = 5. In the cases y> 5, the method of the parameter evolution was implemented.
Using Eqs. (10.8) and (10.11) and setting V = V0, we calculated numerically the GTUNNETT dc characteristics: spatial distributions of the dc electric potential and the dc components of the electron and hole densities, as well as the dc current-voltage characteristics. The pertinent results are shown in Figs. 10.2-10.5.
Figure 10.2 Spatial distributions of electron and hole concentrations (upper panel) and of electric potential (lower panel) in the GTUNNETT i-section.
Figure 10.3 Current-voltage characteristics for GTUNNETTs with different number of GLs K.
Upper panel in Fig. 10.2 shows examples of the spatial distributions of the electron and hole sheet concentrations in the i-section. The spatial distributions of the dc electric potential calculated for GTUNNETTs with different numbers of GLs К at fixed voltage are shown in Fig. 10.2 (lower panel}. One can see that an increase in К results in a marked concentration of the electric field near the doped sections. This is because at larger K, the net tunneling generation rate becomes stronger. This, in turn, results in higher charges of propagating electron and hole components, particularly, near the p-i- and i-n-junctions, respectively.
Figure 10.4 Dependences of dc current on dielectric constant for different numbers of GLs K: net current—upper panel and current per one GL—lower panel.
Figure 10.3 demonstrates a difference in the dc current-voltage characteristics in GTUNNETTs with different numbers of GLs K. As can be seen in Fig. 10.3, the dc current becomes larger with increasing К at more than /(-fold rate. In paricular, Fig. 10.4 (upper panel} shows the three-fold increase in К leads to more than three-fold increase in the dc current particularly at relatively small dielectric constants. This is confirmed by plots in Fig. 10.4 (lower panel), from which it follows that the dc current in each GL is larger in the devices with larger number of GLs. Such a behavior of the dc current-voltage characteristics is attributed to the following. An increase in /Heads to an increase in the dc current not only because of the increase in the number of current channels (this would provide just the /(-fold rise in the dc current) but because of the reinforcement of the self- consistent electric field near the edges of the i-section (as mentioned above) and, hence, strengthening of the tunneling injection.
Figure 10.5 Dependences of dc current on number of GLs К at different voltages.
It is worth noting that when the dielectric constant is sufficiently large (about 15-20), the dc current in one GL is virtually the same in the devices with different number of GLs, because in this case, the charge effect is suppressed. In this case, the value of the dc current (per one GL) is approximately the same as that obtained in the above numerical estimate using analytical formula given by Eq. (10.16). In
GTUNNETTs with a moderate value of a, the charge effect leads to superlinear dependences of the net dc current vs number of GLs as seen in Fig. 10.5.
