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Chapter 3 Nanomorphic electronics > Appendix 2: Derivation of electron travel t... - Pg. 87

Appendix 2: Derivation of electron travel time (EQ. 3.55) 87 h w min w p ffiffiffiffiffiffiffiffiffiffiffi 2 2mE b (A1.7) Note that (A1.7) is close to the Heisenberg distinguishability length a H (3.10) derived for tunneling. Generally speaking, Eq. (A1.5) describes a situation of infinite height of the walls [11], and for a finite barrier height, the solution is obtained numerically (see e.g. [11] for a detailed procedure). However (A1.5) can still be used for order-of-magnitude estimates in the case of low barriers. For example, let E b ¼ k B T , then the effective barrier for a confined particle as a function of the well size is plotted in Figure A1.2 calculated using approximation (A1.5) and by an exact numerical solution [11]. As can be seen, both approaches yield similar result for larger w, e.g. w ~ 10 nm and diverge for smaller w, remaining however within a reasonable range for order-of-magnitude estimates. As an interesting observation, the simple approximation (Eqs. A1.5 and A1.7) suggests w min ~ 4 nm, where the effective barrier for electron becomes very small. On the other hand, the exact solution says that at w ~ 4 nm the effective barrier height is wk B T ln2, i.e. the Boltzmann's limit for the minimum barrier height (3.18). APPENDIX 2: DERIVATION OF ELECTRON TRAVEL TIME (EQ. 3.55) The travel time of the electron along the distance L is determined by electron's average velocity hvi: