To illustrate the application of this irreversible open system model to a specific physical system, let us consider the semiconductor heterostructure resonant-tunneling diode (RTD) (Chang, Esaki, and Tsu, 1974; Sollner et al., 1983). The study of this device provided the original motivation for the present investigation. The RTD exploits the ability of modern heteroepitaxial technologies to grow extremely thin layers of chemically different semiconductors (such as gallium arsenide, GaAs, and aluminum arsenide, AlAs) on top of one another in a single crystal structure. To a surprising degree of accuracy, the effects of such a structure on the motion of free electrons (or holes) may be modeled by an effective potential which is related to the local energy-band gap and is thus a function of the local chemical composition (Dingle, Wiegmann, and Henry, 1974). Therefore, a structure consisting of a layer of GaAs a few nanometers thick placed between layers of AlAs (or more commonly, a solid solution with ) forms a rectangular potential well of finite depth for electrons. The shift in energy due to size quantization of the states in the well is enormously enhanced by the low effective mass of electrons in GaAs (0.067 of the free electron mass), so the same shift is obtained in quantum wells tens of atomic layers thick in GaAs as would be obtained in structures of atomic dimensions in free space.
Figure 10. Summary of the properties of the quantum-well resonant-tunneling diode. The curve of an experimental device (Reed et al., 1989) at a temperature of 77 K is shown to the left. The diagrams to the right show the conduction-band profile of the device at different bias voltages corresponding to the noted points on the curve. The shaded regions show the occupied electron states. In equilibrium (A) the current is zero. As a bias voltage is applied, the resonant level (dotted line) is pulled down in energy so that it lines up with the occupied electron states, permitting resonant tunneling (B). As the voltage is increased, the resonant level eventually passes below the lowest occupied state in the cathode (left-hand electrode), and the resonant-tunneling current ceases (C). The current subsequently increases as conduction through higher-energy states becomes possible. The rise in the conduction-band potential near the quantum well apparent in (A) is the result of a nonuniform distribution of impurity ions which is a part of the design of the device.
The behavior of the resonant-tunneling diode is summarized in Fig. 10. The device consists of a quantum well bounded by barrier layers which are thin enough to permit tunneling. Outside the barrier layers are thick layers of lower effective potential which are doped so as to have a significant density of free electrons and to which electrical contact is made. The confined states in the quantum well thus become resonances in this structure and electrons may readily tunnel through these resonances only if they have the correct energy. The energy of the resonance varies with the externally applied electrostatic potential. In particular, at a sufficiently large bias voltage the resonance is pulled below the lowest occupied state in the cathode layer and the resonant tunneling current ceases. This leads to a decreasing current with increasing voltage (``negative differential resistance''), which is an unambiguous indication of resonant tunneling in this structure.
Over the past few years a great deal of work concerning the resonant-tunneling diode, both theoretical and experimental, has been published. Most of the theoretical treatments are expressed in terms of the transmission probabilities associated with pure quantum states. Due to the volume of this work, no attempt will be made to comprehensively review it here, but we will instead concentrate upon the kinetic models.