ELECTRONIC EXCITATION ENERGY TRANSFER BETWEEN QUASI-ZERO-DIMENSIONAL SYSTEMS

Electronic excitation energy transfer is studied theoretically within a prototype system of quantum dots using the excitonic representation of the electronic states of two quasi-zero dimensional subsystems, between which the excitation energy is transferred in a process treated as an irreversible kinetic phenomenon. The electron-phonon interaction is used to circumvent the energy conservation problem, especially when considering the uphill and downhill excitation transfer processes. The theory is studied upon utilizing a simplified model of two interacting quantum dots, both coupled to their environment. The theoretical approach is documented by numerical calculations. The results could be relevant to various cases of the electronic excitation energy transfer between quasi zero dimensional nanostructures.


INTRODUCTION
The semiconductor quantum dots, reminding sometimes artificial atoms, are promissing nanostructures for various applications like quantum information processing or the light emission nanostructures.In the latter case often the hybrid nanostructures combine nanostructures with suitable light absorption properties and nanostructures having useful light emission property.In such zero-dimensional nanostructures the light is supposed to be absorbed say in one part (A) of the hybrid nanostructure.The electronic excitation energy is then transferred to the other part (B) of the nanostructure, and then the light can be emited from the component B. As an example, A part can be a quantum dot, or another quantum nanoparticle, while B can be an organic molecule.The transfer of the electronic excitation energy from A to B is a kinetic process of the transfer of electronic energy and/or electronic charge and as such it remains to be an interesting irreversible transfer problem, until the present days.It is the purpose of the present work to consider some of the basic questions of the energy/charge transfer inside the hybrid quantum dot nanostructure.

ENERGY TRANSFER PROBLEM
We shall confine our attention to the simple case when the light energy is absorbed in the part A, and then the excitation energy is transferred to the part B of the nanostructure, from where it is emitted as a light.The light emitted from the nanostructure is the photoluminescence response of the original light absorption.We shall deal with the particular case when the energy transfer between A and B is performed with help of the electrostatic coulombic interaction, like the dipole-dipole interaction, of the molecules A and B, as it is the case in the Fӧrster's mechanism of the energy exchange [1].
The energy transfer under study is a nonequilibrium process.Probably in practically all the available literature it is treated in the frame of the perturbation calculation with taking the electronic excitation transfer dipole-dipole interaction as a small perturbation [2].Such formulas can thus be questioned from the point of view of the energy conservation in the course of the electronic excitation energy transfer [3].
We are going to demonstrate a derivation of the energy transfer rate formula based on using the excitonphonon interaction in the parts A and B as a perturbation, with including the latter interaction nonperturbatively.Because of the limited space in this publication we shall confine ourselves to a brief outline of the theory and to a brief display of numerical results.We will show that the presently used nonadiabatic approximation to the inclusion of the electron-phonon coupling into excitonic self-energy can give us an explanation of the experimentally well known energy transfer processes in which the energy is transferred to the part B with the electron excitation energy larger/smaller than is the excitation energy in the part A (the so called uphill or downhill relaxation) [4].

ENERGY TRANSFER
We use the excitonic representation of the electronic system eigenstates of the parts A and B of the hybrid particle.We assume the existence of two exciton states only, one is the exciton localized at the part A, the other exciton is localized at the part B of the hybrid nanostructure.We assume that the purely electronic part of the Hamiltonian of the system consists of the Hamiltonian of free excitons exc H 0 at A and B, and of the dipole-dipole interaction F V which operator is approximated by the exciton transfer hermitian operator quadratic in the exciton particle operators.We include also the exciton-phonon interaction operator ph exc V  which provides the scattering of the exciton on phonons without exciton transfer between A and B.
It is interesting to consider a simple case, in which the values of the exciton energies are different, so that the difference between the exciton energies at the parts A and B, which are respectively , where t is parameter giving the strength of the exciton transfer interaction.
We perform the exact diagonalization of the pure excitonic part of the Hamiltonian, to obtain a new representation of the exciton states in the hybrid parts A and B. In the new representation the excitonphonon interaction ph exc V  now gains a longitudinal part, namely such terms, which transfer the exciton between A and B while simultaneously emitting or absorbing a phonon.

Fig. 1 Exciton energy downhill transfer rate from the part B to A (full) and uphill transfer rate from A to B (dash).
Exciton transfer parameter t=10 meV.The resonances mark integer multiples of the optical phonon energy.The exciton transfer rate is computed as dependent on the separation between magnitudes of exciton energy in the two parts A and B of the hybrid nanostructure.
Performing the above indicated canonical transformation, then under the condition of

 
the system becomes to be formally identical with the earlier studied system of the charge transfer between two 16.-18.10. 2013, Brno, Czech Republic, EU molecules [5,6].The reader is referred to the works [5,6] and to our later publications for a more detailed account of the technical development of the problem of the transfer.
We deal with the energy transfer between the parts A and B of the hybrid particle, under the perturbation, which is proportional to a product of the transfer interaction operator t and the original transverse exciton- phonon interaction operator.Using the new representation of the exciton-phonon operator we develop the quantum kinetic equation which gives the rate of the energy transfer between the parts A and B of the hybrid nanostructure.The new exciton-phonon interaction is included into the exciton self energy in the self-consistent Born approximation (SCB).In the restricted dimensionality of the system of the presently considered inclusion of the exciton-phonon interaction in the SCB approximation can be interpreted as nonadiabatic inclusion of the exciton-phonon coupling [5,6].

NUMERICAL RESULTS AND CONCLUSIONS
In the Fig. 1 the energy transfer rate is calculated for the case of the exciton transfer from part A to part B, or from B to A, when We see that the present (off-shell) theory allows for the energy transfer without any exciton energy restrictions.The explanation of this feature is in that the exciton transfer is made possible by considering the nonadiabatic influence of the electron-phonon interaction in the present quasi-zero dimensional system.The present calculation of the energy transfer is analogical to our earlier demonstration of the electron charge transfer demonstrated theoretically on the example of the electron transfer between two neighboring molecular bases in the DNA molecule [7].
We simulate the energy transfer between A and B by a model, in which A and B are quantum dots with material parameters of GaAs crystal [8].Within the present model the exciton interacts with optical vibrations of GaAs, with phonon energy 36.2 meV.This interaction leads to certain resonances observed in the dependence of the excitation transfer rates demonstrated in Fig. 1, These resonances occur at the integer multiples of the optical phonon energy.
The exciton transfer rate from A to B, in the case when the exciton energy on the B part is larger than in the A part, represents a supporting argument in favor of our view that the uphill and downhill energy transfers observed often experimentally [4] can be interpretted as being due to the inclusion of the Förster's energy transfer mechanism together with the exciton-phonon coupling included nonperturbatively as shown in this work.
In the conclusion, we can say that in the present work we have demonstrated that the inclusion of the exciton interaction with the vibrations of the atomic lattice, in the approximation going beyond the limits of a perturbation calculation of finite degree, makes it possible to explain the experimentally observed uphill and downhill processes of the electronic excitation transfer [4].In this way we may suggest how to circumvent effectively the question of an energy conservation problem sometimes discussed in similar irreversible transfer processes theories [3].Let us remark that a more detailed account of the presently shown theoretical argumentation will be presented elsewhere.
than the exciton transfer operator