Electron tunnelling phase time and dwell time through an associated delta potential barrier

The electron tunnelling phase time \tau^\rm p and dwell time \tau_\rm D through an associated delta potential barrier U(x)=\xi\delta(x) are calculated and both are in the order of 10^-17 -10^-16 s. The results show that the dependence of the phase time on the delta barrier parameter \xi can be described by the characteristic length l_\rm c =\hbar^2/m_\rm e\xi and the characteristic energy E_\rm c=m_\rm e\xi^2/\hbar^2 of the delta barrier, where m_\rm e is the electron mass, l_\rm c and E_\rm c are assumed to be the effective width and height of the delta barrier with l_\rm cE_\rm c=\xi, respectively. It is found that \tau_\rm D reaches its maximum and \tau _\rm D=\tau^\rm p as the energy of the tunnelling electron is equal to E_\rm c/2, i.e. as l_\rm c =\lambda _\rm DB, \lambda_\rm DB is de Broglie wave length of the electron.