Wave Propagation in Linear Media | re-examined

4.6 Examples of tunnelling events
7
6
5
4
3
2
1
τ
0.5 1 1.5 2 2.5 3 Ω
8
6
4
τ
0.5 1 1.5 2 2.5 3 Ω
Figure 4.35: Numerically determined tunnelling times compared with the phase time theory for thin
(D = 1, left graph) and medium-sized (D = 5, right graph) barriers and a small-band Gaussian
electron ( = 100, n = 4).
20
15
10
5
-5
-10
τ
0.5 1 1.5 2 2.5 3 Ω
20
15
10
5
τ
0.5 1 1.5 2 2.5 3 Ω
Figure 4.36: Numerically determined tunnelling times compared with the phase time theory for a thick
barrier (D = 10), but electrons of di erent bandwidths (left graph: =10:54 b= k = 6 at = 0:2,
right graph: = 35:12 b= k = 20 at = 0:2, n = 4 in both cases). The left plot also shows the
semi-classical time (dashed line).
Fig. 4.35 shows the results for a small-band electron and two di erent barrier widths. Out of all
theoretical tunnelling time approaches in section 4.5, the phase time suits best the numerical
results. This is not really surprising, because the way we set up the numerical computation of
the tunnelling time mimics exactly the basic idea of the phase time. We follow the motion of
the wave packet's maximum, and furthermore the spectrum is sharply peaked about a single
wave number. Note that for low incident energies, the phase time prediction and the numerical
results disagree. Collins et al. attributed this behaviour to the transmission coe cient they
thought to be exponentially varying at low energies | an argument we cannot follow after
inspection of g. 4.18, which shows the transmission coe cient to be particularly smooth for
small = p !=!p. Rather, we can identify the inevitable pulse reshaping phenomenon as
the reason of this discrepancy. Owing to our assumptions, the spectral width of the wave
packet is held constant irrespective of its centre frequency. Hence the extent to which the
frequency can be shifted during the tunnelling event is also xed or at least upper-bounded.
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