3.6 Bound and Quasi-Bound States

Up to now it has been assumed that all energetic states in the substrate contribute to the tunneling current. However, the high doping and the high electric field in the channel leads to a quantum-mechanical quantization of carriers as described in Section 2.2.1 [156,157]. If it is assumed that the wave function does not penetrate into the gate, discrete energy levels can be identified. However, it cannot be assumed that electrons tunnel from these energies, since for the derivation of the levels it was assumed that there is no wave function penetration into the dielectric. This leads to the paradox which was addressed by MAGNUS and SCHOENMAKER [158]: How can a bound state, which has vanishing current density, lead to tunneling current?

The answer is that it cannot. Taking a closer look at the conduction band edge of a MOSFET in inversion reveals that, depending on the boundary conditions, different types of quantized energy levels must be distinguished [159], see Fig. 3.13: Bound states are formed at energies for which the wave function decays to zero at both sides. Quasi-bound states (QBS) have closed boundary conditions at one side and open boundary conditions at the other side. Free states, finally, are states which do not decay at any side. The total tunnel current density therefore consists of current from the QBS and from the free states:

$$J=\ensuremath {\mathrm{q}}\sum_i \frac{\ensuremath{n_\nu}({\mathc... ...}} TC({\mathcal{E}}) N({\mathcal{E}})\,\ensuremath {\mathrm{d}}{\mathcal{E}}\ ,$$ (3.92)

where the symbol $\ensuremath{n_\nu}({\mathcal{E}}_i)$ denotes the two-dimensional carrier concentration [160]

$$\ensuremath{n_\nu}= \ensuremath{g_\nu}\frac{m{\mathrm{k_B}}T}{\pi... ...\mathcal{E}}_\mathrm{f}}- {\mathcal{E}}_i}{{\mathrm{k_B}}T} \right) \right) \ ,$$ (3.93)

the symbol $g_\nu$ is the valley degeneracy, and $\ensuremath{\tau_{\mathrm{q}}}$ is the life time of the quasi-bound state ${\mathcal{E}}_i$. The life time is based on GAMOW's theory of nuclear decay [40] and denotes the time constant with which an electron leaks through the energy barrier. Since bound and quasi-bound states are closely related, the computation of bound states will be described first.

Figure 3.13: Free, bound, and quasi-bound states in a typical MOS inversion layer.

Subsections

• 3.6.1 Eigenvalues of a Triangular Energy Well
• 3.6.2 Eigenvalues of Arbitrary Energy Wells
• 3.6.3 The Life Time of Quasi-Bound States
  • 3.6.3.1 The Reflection Coefficient Resonances
  • 3.6.3.2 The Quasi-Classical Formula
  • 3.6.3.3 The Eigenvalues of the Non-HERMITian HAMILTONian

A. Gehring: Simulation of Tunneling in Semiconductor Devices