The effect of the Coulomb charging energy on the time evolution of the electron wave packet in a coupledquantum-dot structure is investigated by use of coupled equations derived from the Schrodinger equation. As long as the Coulomb charging energy does not exceed a certain threshold energy (four times as much as the coupling energy between the dots), the electron wave packet, which was initially localized in one of the dots, is completely transferred to the other dot and oscillates back and forth between the two dots. If the Coulomb charging energy exceeds the threshold, the transfer rate of the wave packet becomes abruptly incomplete and is reduced to less than one-half. This effect may be called the Coulomb blockade of the resonant electron tunneLing in coupled quantum dots. It should be stressed that the transfer period of the wave packet is remarkably increased at and near the threshold charging energy. The analysis presented here should provide a useful basis for the design and evaluation of mesoscopic devices operating with coupled quantum structures.One of the most fundamental problems of quantum mechanics is the time evolution of a nonstationary wave packet. ' The motion of such a wave packet is purely coherent if phase-breaking interactions with a thermal bath are absent. Experimentally, we can observe such coherent quanturn dynamics if the dimension is comparable to or less than the mean free path of the electrons. Owing to the recent advances of semiconductor technologies, it has been possible to reduce the critical dimensions of devices to below the mean free path of the electrons and to increase the mean free path itself by suppressing scattering and using a smallelectron effective-mass material.In this paper, we investigate the effects of the Coulomb charging energy ' on the electron tunneling in a coupledquantum-dot structure. First, we briefly review the time evolution of the electron wave packet in the coupled-quantumdot structure without considering the Coulomb charging energy. ' ' If an electron is initially in one of the two dots, its wave function is given as a linear combination of the symmetric (energy E, ) and antisymmetric state (energy E, ) with equal amplitudes. Each state is associated with a timedependent phase factor of the form exp( -iE, , t/6). Because the two states have the energy difference BE=E, -F" their phase factors advance at different rates, and the resultant linear combination wave function corresponds to an electron wave packet oscillating back and forth between the two coupled dots with a frequency f = SE/h. If the electron is in one of the two dots at t =0, it will be found in the other dot after one-half period of this oscillation, that is, after the time tr = h/2bE, which may be considered as the time required for an electron to transfer to the other dot. The possibility to observe electron oscillations in solids has been discussed extensively in the literature. Recently, strong evidence for electron oscillations in a coupledquanturn-well structure from optical pump-...
The effect of the Coulomb charging energy on the time evolution of the electron occupation probability in a coupled-dot system is numerically investigated by use of nonlinear coupled equations derived from the Schrodinger equation. For the initial condition that the electron occupation probability is equally distributed to two quantum dots, a small initial fluctuation in the distribution of the electron occupation probability causes the symmetry-breaking instabilities of the distribution for Coulomb charging energies larger than a certain threshold value. The numerical analyses with various initial conditions are also discussed.One of the most fundamental problems of quantum mechanics is the time evolution of a nonstationary wave packet.Bavli and Metieu have shown that a semi-infinite laser pulse can be used to localize an electron in one of the wells of a double-well quantum structure. Also Grossman et al. have pointed out an interesting effect of a cw laser acting on an electron in a double well. If the electron is initially localized in one of the wells, and if the laser power and frequency are chosen appropriately, the radiation field can prevent the electron from tunneling back and forth between the wells. In our previous paper we have shown that by introducing Coulomb charging energies into the conventional coupled equations, the equations become nonlinear and the Coulomb charging energy strongly modifies the electron oscillations between dots: The electron can completely tunnel from one of the dots to another for small Coulomb charging energies, while for the charging energies exceeding a certain threshold, the electron tunneling becomes abruptly incomplete with the transfer rates less than one-half. This effect has been called the Coulomb blockade for the coherent oscillations of the electron (or Coulomb blockade for the quantum beat).In this paper, we numerically investigate the effects of the Coulomb charging energy on the electron tunneling in a coupled-dot structure with different initial conditions from those in our previous work. We are especially interested in an initial condition in which an electron is equally distributed in the two dots; in other words, the electron is initially in the lowest eigenstate of the coupled-dot system. The time evolution of the wave function is treated by the time-dependent Schrodinger equation: d where H is a Hamiltonian. We express an eigenstate of an electron in the coupled-dot system as a linear combination of the wave function t/ll (j = a, b) of the isolated quantum dots a and b, i.e. , Q(t) =a(t) P. +b(t) At, .(Without considering the Coulomb charging energy, substituting Eq.(2) into Eq. (4), we obtain a simple linear coupled equation such as Eq. (7) in Ref. 9. Introducing the Coulomb charging effects into the linear coupled equations as the phase-mismatching term, the resultant form becomes nonlinear as follows: da dt =iA(1 -2IaI )a -i trb, (3a) db dt = -iA(1 -2IaI )b -isa,where a and b are complex normalized amplitudes given in Eq.(2) (IaI +IbI =1), A=e /4Ac, ...
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