Another aspect of the behavior of electronic devices which is of much interest to circuit designers is the small-signal ac response of the device. This is the response of the device to a small sinusoidal voltage imposed upon a generally much larger dc bias voltage. That is, one seeks to evaluate the effect of a small perturbation on a far-from-equilibrium steady state. This is a rather different problem from that treated by the linear response theory of statistical physics (Kubo, 1957), which seeks to evaluate the effect of small perturbations on an equilibrium state. A perturbation expansion of the present kinetic theory may be readily obtained to evaluate the small-signal ac response of our model RTD (Frensley, 1987b, 1988a; Mains and Haddad, 1988b). Let us assume that the potential of the system varies as
where denotes the complex conjugate, is the dc potential including the heterostructure and the large bias voltage, is the potential due to the small ac voltage, and is a perturbation parameter introduced solely to keep track of the order of the perturbation (and is ultimately set equal to unity). We should expect that the current induced in the external circuit can be expanded as
where and are the total voltages applied, e being the charge of the electron. The coefficients of (5.61) describe different aspects of the ac response: y is the linear admittance, the amount of rectification of the sinusoidal waveform is given by , and the amount of second-harmonic generation is given by . Note that at these coefficients are just the derivatives of the curve: and . The coefficients of (5.61) at an arbitrary frequency may be obtained from the corresponding components of the Wigner function. To do this we write the Liouville operator as
The Wigner function can be expanded (to second order in ) as
Inserting (5.62) and (5.63) into the Liouville equation and collecting terms of equal frequency and order in leads to these equations:
where is obtained from (4.56). (The denominators of this perturbation series look a bit unfamiliar, with expressions of the form rather than . The reason for this is that we have mixed the quantum-mechanical convention for the time dependence, , with the convention used in electronics, . While a consistently quantum-mechanical notation would produce more conventional expressions, it would also produce a great deal of confusion when we examine the imaginary parts of the response to determine whether they resemble capacitances or inductances.) The superoperator resolvent expressions in (5.64)--(5.66) are readily evaluated with the same algorithms used to solve the steady-state and transient problems.
Evaluating the expectation value of the current density J for any of the terms of gives the conduction current as a function of position q:
The current induced in the external circuit by this conduction current within the device is obtained by invoking the Shockley-Ramo theorem (Shockley, 1938; Ramo, 1939). We will approximate the properties of the doped contacting layers as ideally metallic conductors bounded by interfaces to the higher-potential barrier layers at and . The Shockley-Ramo theorem then takes the form:
where A is the area of the device. The coefficients of the expansion of (5.61) are thus given by:
It should be emphasized that these expressions represent only the conduction current component; the displacement current must be added to them to obtain a complete description of the behavior of the device.
Figure 17. Small-signal ac response of the resonant-tunneling diode for a dc bias of 0.17 V, which places the device in the middle of the negative-resistance region. The device conductance (the real part of the admittance, solid line) is negative at lower frequencies, with a value equal to that expected from the derivative of the dc curve. The negative conductance decreases in magnitude and becomes positive at a few THz. The complex behavior at higher frequencies is an indication that optical transitions are becoming important. The susceptance (imaginary part of the admittance, dashed curve) has the same sign as a capacitance and is due to the effects of electron storage in the quantum well. These quantities reflect only the conduction current and do not include the displacement current through the parasitic capacitance of a real device. This displacement current would prevent observation of the higher-frequency effects in a realistic experimental situation.
The linear admittance y of the present RTD model was evaluated using (5.64) and (5.70) at a bias of 0.17 V (in the middle of the negative-resistance region), as a function of frequency over the GHz and THz regions. The results are plotted in Fig. 17. The conductance is negative at lower frequencies, as we would expect from the dc results. This negative conductance ``rolls off'' and becomes positive at about 6 THz, which is therefore the maximum frequency of oscillation of the intrinsic device (not including parasitic effects). The susceptance is positive and proportional to at lower frequencies, which is the behavior of a capacitance. Recall, however, that the displacement current which flows through the geometrical device capacitance is not included in this calculation. The result that is somewhat surprising, since the most obvious reactive effect in electron transport at high frequencies is the electron inertia, which leads to resembling that of an inductor with negative (Champlin, Armstrong, and Gunderson, 1964). The initial calculations of the admittance by the present author (Frensley, 1987b, 1988a) gave negative due to a programming error, and the electron-inertia explanation was proposed in those papers. During the preparation of the present work the error was discovered, and correcting it brings the results into agreement with those obtained by Mains and Haddad (1988b), who obtained positive . Thus, the electron inertia does not explain the behavior of , and an alternative explanation must be sought. A key piece of evidence is provided by evaluating the admittance of structures with either one energy barrier or none, in addition to the double-barrier structure. These structures do indeed show negative (inductive) , presumably due to electron inertia. The capacitive is thus uniquely associated with the double-barrier structure and therefore must reflect the confinement of electrons in the quantum well. The idea that electron storage in a quantum well could be represented as a capacitance was proposed by Luryi (1985), but he identified this capacitance with the geometrical capacitance of the device, through which the displacement current flows. The storage capacitance inferred from the present calculation is 1--2 orders of magnitude smaller than the geometrical capacitance.
Figure 18. Nonlinear response of the resonant-tunneling diode at a dc bias of 0.13 V, at the peak of the curve. The rectification coefficient (solid line) shows a resonant enhancement near 6 THz.
The rectification and second-harmonic generation coefficients and were evaluated using (5.65), (5.66), (5.71), and (5.72) at a bias of 0.13 V (the top of the current peak). The moduli of these quantities are shown in Fig. 18. While decreases at higher frequencies, shows a resonant enhancement over the frequency range of 1 to 8 THz. This is quite interesting, because was measured by Sollner et al. (1983) at a frequency of 2.5 THz. The experimental data show that for most bias voltages exceeds the dc , indicating that the magnitude of must increase in this frequency range. On the other hand, the rectification process in the RTD has been recently analyzed by Wingreen (1990), using a transmission-coefficient approach. He found no evidence of enhancement, only a decrease in as the frequency is raised. One difference between Wingreen's calculation and that based upon (5.65) is that the former includes the effects of only one resonant level, whereas the latter includes all such levels. This suggests that the enhancement of might involve transitions between resonant levels, though the frequency of the transition between the lowest two levels in the present example is 60 THz, which argues against this notion. This illustrates one of the problems with a kinetic approach which incorporates all physical processes: Such an approach provides little guidance when one desires to identify that process which is the cause of some particular effect.
Figure 19. Linear component of the ac current density (divided by the applied ac voltage and thus expressed as an admittance) as a function of frequency and position. At lower frequencies the current density is spatially uniform, but strong nonlocal effects develop as the frequency is increased. This is a characteristic of the transition from electronic to optical behavior. The prominent peak in centered in the quantum well at 50 THz is due to quantum transitions between the two lowest resonant levels.
It is particularly interesting to look at as a function of both frequency and position q. This is plotted in Fig. 19. At frequencies below a few THz the current is independent of position, as one would expect in an electron device. As the frequency increases above this value, the ac current density becomes strongly nonuniform, indicating that the response of the current to the applied potential is strongly nonlocal. A particularly prominent peak occurs in at a frequency of 50 THz and centered within the quantum well. The positive value of the conductance in this peak indicates that the in-phase current density is locally large, so this part of the device is absorbing power from the ac electric field. The obvious explanation for this absorption is that the peak reflects quantum transitions between the two lowest resonances in the well. A transmission-coefficient calculation indicates that, for the present example, these states are separated in energy by 0.248 eV, for which the corresponding photon frequency is 60 THz. The small discrepancy in predicted frequencies is presumably attributable to the effect of the Markov assumption in the kinetic theory, as in the case of the curves. Figure 19 is interesting because it gives us a view of the transition of a single system from the domain of electronics to that of optics.
In addition to the these effects, the irreversible open-system models have been applied to investigations of the effects of phonon scattering, as described in Appendix 14, and the self-consistent potential in the RTD, as described in Appendix 9. The various applications of open-system kinetic theory to RTDs clearly demonstrate the value of this approach, in spite of the existence of several unresolved mathematical issues which will be explored in the next section.