2.4 Tunneling

  In 1923 L. de Broglie [24] introduced a new fundamental hypothesis that particles may also have the characteristics of waves. Schrödinger expressed this hypothesis 1926 in a definite form which is now known as the Schrödinger wave equation. The continuous nonzero nature of its solution, the wave function which represents an electron or particle, implies an ability to penetrate classically forbidden regions and a probability of tunneling from one classically allowed region to another. Fowler and Nordheim [33] explained in 1928 the main features of electron emission from cold metals by high external electric fields on the basis of tunneling through a triangular potential barrier. Conclusive experimental evidence for tunneling was found by L. Esaki in 1957 [26] and by I. Giaever in 1960. Esaki's tunnel diode had a large impact on the physics of semiconductors, leading to important developments such as the tunneling spectroscopy, and to increased understanding of tunneling phenomena in solids. L. Esaki [27], I. Giaever [41] and B. Josephson [59] received 1973 the Nobel prize for their work about tunneling in semiconductors, superconductors and theoretical predictions of the properties of a supercurrent through a tunnel barrier, respectively. The concept of resonant tunneling in double barriers was first introduced by R. Davis and H. Hosack [21]. At about the same time C. Neugebauer and M. Webb [90] and some years later H. Zeller and I. Giaever [110] and J. Lambe and R. Jaklevic [73] studied granular films. They observed a current suppression at low bias voltage, which is today known as the Coulomb blockade. It took almost two decades until in 1985 D. Averin and K. Likharev [8] formulated the  `orthodox' theory of single-electron tunneling or short SET, which quantitatively describes important charging effects such as Coulomb blockade and Coulomb oscillations. The orthodox theory makes the following assumptions: It neglects dimensions and shapes of islands. Thus it is a zero-dimensional model. The tunnel process is assumed to be instantaneous. Actually the  tunnel time, the duration an electron spends below a barrier, is of the magnitude of 10^-14 s [47]. Charge redistributions after a tunnel event are also assumed to be instantaneous. In addition  energy spectra in leads and islands are taken to be continuous. The main result of the orthodox theory is that the rate of a tunnel event strongly depends on the change in free energy the event causes.