Quantum tunneling time

We present a calculation of quantum tunneling time based on the transition duration of wave peak from one side of a barrier to the other. In our formulation, the tunneling time comprises a real and an imaginary part. The real part is an extension of the phase tunneling time with quantum corrections whereas the imaginary time is associated with energy derivatives of the probability amplitudes.

Figure 1

A comparison of numerical traversal times for a thin barrier (width $25{\AA }$, height $0.1\text{eV}$, effective mass $m^{*}=0.063m_e$), where $t$ is in unit of femtosecond. The solid line is our result, the dotted line denotes Büttiker’s expression, and the dashed line is the result for phase time.

Figure 2

Tunneling time as Fig. 1 for a thick barrier (width $110{\AA }$, height $0.2\text{eV}$, effective mass $m^{*}=0.063m_e$).

Figure 3

Same as Fig. 1 for a thick barrier (width $200{\AA }$, height $0.3\text{eV}$, effective mass $m^{*}=0.063m_e$).

Figure 4

A comparison of numerical traversal times for a very small effective mass and a thick barrier (width $200{\AA }$, height $0.3\text{eV}$, effective mass $m^{*}=0.00063m_e$), where $t$ is in units of femtoseconds. The solid line is our result, the dotted line is Büttiker’s expression, and the dashed line is phase time.

Figure 5

Same as Fig. 4 for a thin barrier (width $25{\AA }$, height $0.1\text{eV}$, effective mass $m^{*}=0.00063m_e$).